What are the intellectual roots of the "French Republican identity" pursuit?

One possible answer is that France has always, historically, been a state-prone centralized country, with a unity imposed top-down (Nora, 2010). (It doesn't go in contradiction with totalMongot's answer, both might complete each other).

It is true that the ideology of the "one and unique" France was launched by and during the 3rd Republic (1870-1940), together with its many symbols, principles and laws like the laïcité (French secularism) and the free, mandatory and secular Republican school, enacted by Center-left Republicans, founders of this Republic, such as Jules Ferry and Léon Gambetta, so-called "moderate Republicans". The goal was to build a collective identity stemmed on the Revolution, and at the same time being moderate enough in order to ensure sustainability of this new Republic (after successive failures of the two previous Republics):

When the Republicans came to power at the end of the 1870s, they were faced with a dual task: on the one hand, to accentuate the conflict in order to emerge from the ambiguity and make their vision of French identity as a political collectivity born of the Revolution; on the other hand, to be a "tempered Republic" (Ferry, in Girardet, 1985, p. 250) in order to be able to establish itself over time. This ambivalence explains the internal tensions of French republicanism, particularly between heirs of the Jacobin tradition and anti-Jacobins (Furet, 1986).

(Auberot, 2011)

But it nonetheless already existed centuries before.

History of the unified France

The centralization and the cultural unification of the country as imposed by the rulers started during the Middle Ages and continued during the reign of Louis XIV (newspapers article, written by Laurence Moreau, published by "Le Point"):

Once removed, the monarchical principle has continued to permeate the relationship of the French to power to this day. The president of the Fifth Republic, even though he is elected by the people for a limited period of time, has such power that one could speak of a "republican monarchy." Centralization was not born with the Jacobins and the Terror, but was initiated by Cardinal Richelieu in the sixteenth century and then reinforced by Louis XIV. The aspiration to create a country unified by language, religion and morals did not emerge during the revolution, but as early as the Middle Ages. In the ninth century, the Strasbourg oath established the principle of "one country, one language". In the 16th century, the Edict of Villers-Cotterêts ousted Latin from the administration in favor of French and local languages. Under Louis XIV, French was already the language of power and prestige.

And indeed, the traces of a well-defined identity are already found in the historiography of the 16th century, and is backed by a true dynastic, administrative, and territorial long continuity.

From the XVIth century, at the time of the wars of Religion where the first form of a history of France is constituted, one could find all the major themes which will not leave it any more: the French exceptionality, the French antiquity, the French unity...

(Nora, 2010)

During the years 1792-1793, the 1st Republic (1792-1804) reinforced the idea of a unified country:

It is at this moment that the whole symbolism of unity is constituted. The Salut Public ("Public Salvation"), the "fatherland in danger" have for example stimulated this already well anchored legal need to guarantee the unity of the nation, this autarkic reflex of "alone against all", on which rests a lot of the national imaginary.

(Nora, 2010)

In the "Report on the necessity and the means to annihilate the patois and to universalize the use of the French language", presented at the National Convention of 4 June 1794, Henri Grégoire, member of the Comité d'Instruction Public wrote about the necessity to centralize the country culturally, through the so-called "language of freedom", French, in replacement of all the dialects (30 dialects at the time, when 0.1% of the French inhabitants spoke French):

One can standardize the language of a great nation... This undertaking, which has not been fully executed among any people, is worthy of the French people, who centralize all the branches of social organization and who must be jealous to consecrate as soon as possible, in a Republic that is one and indivisible, the unique and invariable use of the language of freedom.

This policy of cultural h*m*genization, which was applied to the colonies and later since the 1950s is applied to the immigrants, is done under the principle of "universality" of the Republican principles (the ones written in the 1789 Declaration and the French secularism laïcité), of which the French Republic is itself the defender and the French nation the incarnation (Streiff-Fénart, 2009).

The 2nd Republic (1848-1852) adopts as official motto in 1848, because of the Republican Socialist Louis Blanc, "Liberty, Equality, Fraternity", with "Fraternity" added to the principles "liberty", and "equality", since only the two latters existed in the French Declaration of 1789. The theorization of this three-part motto only appeared at the end of the Montagnard period of the Revolution, and was theorized by Robespierre.

During the 3rd Republic, the three-part motto "Liberty, Equality, Fraternity" is adopted as an official symbol of the Republic, and on July 14, 1880 it appears on the pediments of all public institutions.

The one and unique France was reaffirmed by the establishment of the 5th Republic (1958-today), founded by De Gaulle, which by the way limited the power of the parliament and reinforced the power of the executive (the President, his/her Prime minister, and the other ministers). The first article of its Constitution (1958) makes the top-down unification of the country very clear, "indivisible" is the very first adjective modifying the word "Republic":

Article 1: France is an indivisible, secular, democratic and social Republic.

An adapted way of understanding the "French Republican identity" is to perceive it as part of the larger modèle républicain ("Republican model").

Analysis of today's "Republican Model"

The "Republican Model" is made of several components: the idea of "individual-citizen", the French free, mandatory and secular school, and the laïcité (French secularism).

• Individual-citizen

In France, the state has a direct relation with the so-called individu-citoyen ("individual-citizen") (Streiff-Fénart, 2009). This idea of "individual-citizen", a very politicized member of a society, is found in Rousseau (The Social Contract, 1762):

The citizen is an eminently political being (the city) who expresses not his individual interest but the general interest. This general interest is not limited to the sum of the particular wills but exceeds it.

It started from the French Revolution in 1789, which is key in the creation of the citoyenneté ("citizenship"):

Partly anticipated by the thinkers of the Enlightenment and then completed according to the situation, a process of revolutionary acculturation is put in place that transforms the subject into a citizen and, sometimes, into a female-citizen. (...) To be a citizen, it is necessary to participate in the life of the Nation, therefore: to vote, to pay its taxes, to fight for its defense and its security, external or internal. To these conditions of exercise will not be long to be added cultural behaviors: a language and new clothes prohibiting the old marks of social hierarchy and proclaiming the patriotism and the attachment of the citizen to the new values, the assistance to revolutionary festivals to the ritual at the same time civic and educational.

(Duprat, 2015)

Robespierre played a role in the theorization of these Republican festivals and saw as instrument for instruction (Duprat, 2013). Robespierre also defended that the education of children should be communal and forged on the model of the republic of Sparta, to which Henri Grégoire opposed (Plötner, 2003).

Today's French citizenship is defined this way by the political scientist Olivier Ihl:

The French tradition is an abstract conception of citizenship which consists in considering all citizens beyond their singularity, in the framework of a unitary state.

• The school

The school in France is a vector of the French Republican identity, and had at the beginning a military dimension defended by Léon Gambetta, which later gradually eroded, to reappear recently in the form of the Universal National Service:

The second consequence of French laïcité, of even greater magnitude, is that the republican civil religion has established a link between the Enlightenment, reason, democracy, and education, which in the final analysis makes primary instruction the basis of the essential national identity. No country has put so much of itself into school.

(Nora, 2010)

• Laïcité (French secularism)

As alluded in the quotation, the concept of laïcité (French secularism) is by itself a fundamental vector of the French identity.

Secularism is above all a "civil religion", the one that the republican state has constructed both as the cement of its legitimacy and of national identity. Although secularism is constitutionally defined by the principles of neutrality, separation and freedom of conscience, the sum of these principles does not account for the formidable symbolic power that underlies secularism as a civic identity.

(Chelini-Pont, 2005)

Conclusion

Although many of the ways today's France is unified can be traced back in the theorization and action of the rousseauist tradition and its followers (this answer mentioned the Montagnard Robespierre, and the socialist Louis Blanc), embodied in the "Fraternity" value of the Republic's official motto, it would be partial to see the actual "French Republican identity" as the sole oeuvre of rousseauism. Indeed, other actors who participated in its implementation were not rousseauists (we mentioned Henri Grégoire and De Gaulle, and center-left Republicans Ferry and Gambetta). That is not to mention the influence of Ernest Renan, of whom ideas now belong to the French right-wing nationalist tradition. Therefore, the "French Republican identity" would be better seen, as stated at the beginning of this answer, in continuation of a centuries-long history of top-down unification of the country, going beyond the succession of the political regimes and polical ideologies.

Response to comments

"Your quote on Henri Grégoire Report of 4 June 1794 completely undermines the entire thesis of this answer. That he even need to speak on the topic itself speaks to the absence of such a cultural and linguistic unity leading up to 1794."

My answer, as my question, does not talk about actual organic unity of the French people, but about the efforts to create one common unique identity.

References

Auberot, J. (2011). Identité nationale et laïcité en France. Revue japonaise de didactique du français, 6(2), 9-25.

Chelini-Pont, B. (2005). Laïcités française et américaine en miroir. Cahiers de la recherche sur les droits fondamentaux, (4), 107-118.

Duprat, A. (2015). Citoyenneté et régénération (1789-1794). Parlement[s], 22(3), 49-56.

Nora, P. (2010). Les avatars de l’identité française. le Débat, (2), 4-20.

Plötner (2003). De l’abbé Grégoire à la Réunion de l’Ouest, deux approches républicaines de la vulgarisation des savoirs. In Andries (ed). Le Partage des savoirs xviiie-xixe siècles (pp. 163-180). Lyon: Presses Universitaires de Lyon

Streiff-Fénart, J. (2009). Le “modèle républicain” et ses Autres: construction et évolution des catégories de l’altérité en France. Migration Société, (2), 215-236.