Does a classic Trinitarian formulation consider God to be his own "Family"?

I have not read anyone use an idea like a multi-member family of the Trinity. Father, Son and Spirit are indeed three separate Persons, with their own consciousness, wills etc, but They are of one Essence. The Essence is not a Person.

As for Father/Son, the Son was begotton of the Father as to His Sonship (i.e. His role), not begotten as to His Godhood, for in that respect the 3 Persons stand equally as to power and glory.

The idea of family, as an illustration of the 3 Persons Who are God, is not one that is used by the church fathers, that I am aware of.

(Being made in his image I gather (from the Westminster Shorter Catechism) is to be made in knowledge, righteousness and holiness).

Scope: This post offers an answer for the first part of the question, as I understand it: whether or not Jesus' identification of himself as the Son of God makes the Trinity an instance of Partialism.

The Short Answer

No, it doesn't. The fact that there is no earthly analogue for the Trinity makes it difficult to accurately describe in human language, so a certain amount of "lexical flexibility" is necessary, and theological terminology is necessary if one wants to describe the Trinity with any precision. History has affirmed this necessity. Let me try to explain...

The More Detailed Answer

The First Ecumenical Council

The doctrine of the Trinity began to acquire some of the distinctions by which it is known today at the First Ecumenical Council, or Council of Nicea, in 325 CE, though it did exist prior to the fourth century. At the Council, it was established as the orthodox teaching that the Son and the Father were 'of the same substance' (Gk. h*m*ousios), contra the Arian teaching that the Son was created by the Father.

Athanasius of Alexandria

Athanasius was present at the First Ecumenical Council as a young man, and went on to dedicate much of his life to leading the battle against the Arian heresy. In so doing, he contributed much of what we today recognize as the doctrine of the Trinity. Church historian Justo Gonzalez offers the following summary of the doctrinal position at which Athanasius' arrived after many years of study and debate:¹

Finally, in a synod gathered in Alexandria in 362 CE, Athanasius and his followers declared that it was acceptable to refer to the Father, Son, and Holy Spirit as "of one substance" as long as this was not understood as obliterating the distinction among the three, and that it was also legitimate to speak of "three substances" as long as this was not understood as if there were three gods.

The Second Ecumenical Council

Several years later, in 381 CE, the Second Ecumenical Council was held, this time in the city of Constantinople. Its findings were significantly influenced by the work of the Cappadocian Fathers: Basil the Great, Gregory of Nyssa, and Gregory of Nazianzus. The Council affirmed the Nicene faith as orthodox under Emperor Theodosius, and equality of the Holy Spirit was incorporated in the teaching on the Trinity. The council also further clarified the nature of the Trinity. To again quote Gonzalez:²

[The Cappadocian Fathers'] main contribution was in clarifying the difference between ousia ("essence") and hypostasis -- a word that literally means "substance" but which the Cappadocians defined as the translation of the Latin persona. Thus, the Latin West and the Greek East came to agree on a common formula: one essense -- or ousia -- in three persons -- or hypostases.

Conclusion

So, it seems like those are a few important events in Church history that elaborate on some of the complexities involved in attempting to accurately describe the nature of the co-existence of the Father, Son, and Holy Spirit. If the historic teachings are accurate understandings of the Biblical text, then it should be entirely acceptable to understand the Trinity as consisting of multiple "persons" -- so long as we also understand that they are of one "essence" or "substance."

I also think it's worth noting that, while composing a response to another recently asked question about Partialism (at the risk of self-promotion, the question can be found here), I was unable to locate any historic references to the heresy under the particular name, "Partialism." After checking several of my usual primary sources, I went through the first half-dozen or so results pages on a few different Google searches looking for some historical background for the term, all to no avail. As far as I can tell, the source of most of the Google-indexed references to Partialism can be traced to one of two web pages: a Monergism list of Trinitarian heresies, and a comical YouTube video that attempts to explain the Trinity through a cartoon of St. Patrick. Neither one lists any sources or offers historical background (though I've not viewed the entire YouTube video). The next step is to visit a fairly heavy duty seminary library that I thankfully have access to here in town.

Sources:

1: The Story of Christianity, Revised Edition, by Justo Gonzalez. Chapter 19, "Athanasius of Alexandria."

2: The Story of Christianity, Revised Edition, by Justo Gonzalez. Chapter 20, "The Great Cappadocians."

Edit: Further searching has not revealed any historic theological references to the idea of the Trinity as a family. While the two following articles are no exception in that respect, and are from popular, rather than academic sources, you may find them to be of some use, if for no other reason than that they could point the way for continued research.

Note: I'm not familiar with the work of these authors, and can't vouch for the reliability of either source.

* The Father, the Son, and the Holy Spirit: The Trinity as Theological Foundation for Family Ministry, by Bruce A. Ware. Blog of The Center for Christian Family Ministry at Southern Baptist Theological Seminary. Link.

* The Most Holy Trinity: Supreme Model for Family And Marriage, by F. K. Bartels. Article at Catholic.com. Link.