Russian low cost airline Pobeda, an Aeroflot subsidiary, had a policy to explicitly disallow taking a different, un-assigned seat or swapping with other passengers.

So the answer is "not always".

It may or may not work, YMMV. Sometimes they keep these seats reserved for crew rest, sometimes they explicitly want folks to sit in specific places for weight distribution. It's also something attendants need for safety purposes - if you don't sit in a seat assigned to you, they may not be able to identify your remains In case of a crash.

I've seen attendants insisting on assigned seats, I've also seen them allowing moving seats after take off, and I've seen them not checking too.

TL;DR: Yes, but…

Moving to a different service class (from economy to premium economy, business or first for instance) is just a no-no if you weren’t invited to do it.

Note that in some cases there are slight differences between seats (e.g. more legroom for the first few rows, or the middle seat never being sold in the first few rows) which count as different “classes” with no obvious separation (or a very flimsy one).

Some airlines will also refuse that you move to a seat they would usually sell for a higher price even though it’s physically identical (for instance the first few rows because people who are in a hurry prefer them, emergency exit rows because there’s often more legroom, etc. — see below for more about emergency exit rows).

On some planes, depending on the number of people on board, moving may involve issues with weight distribution (mostly smaller planes with few passengers on board). This is most important during take-off, so crew may tell you off if you try to move before take-off. After that it’s usually fine, though I believe I remember some cases where crew insisted on people getting back to their assigned seat for landing (no idea what the actual reason for this could be, though).

On long flights, some seats are reserved for crew rest, though it’s usually quite obvious in my experience (and on more recent aircraft there is a dedicated space for this above or below the main cabin).

Another possible exception relates to seats in emergency exit rows: they need to ascertain that anybody seating there is an adult, able to understand crew orders and communicate clearly, able to open the exit (especially in the case of over-wing exits, those can be heavy), and in some cases they will even ask confirmation that the passenger is willing to operate the exit. I think there are airlines where you have to actually confirm you meet the requirements and agree to perform the tasks involved before you can even book those seats.

Another point to note about emergency exit rows: it is often the case that seats in the row in front of such a row do not recline. Check Seat Guru for the specifics on each aircraft.

Yet another exception is that a seat may not be usable because of a technical issue: it may just be a comfort issue (e.g. seat does not recline, tray table is broken, IFE does not work…), or it may be a safety issue (seat belt missing or broken, life vest missing, issue with the emergency oxygen supply…). This is quite the exception, but it still happens regularly.

I’m not quite sure you can rely on seat maps for finding an available seat at this stage though. At some point during the check-in window the aircraft will usually be transferred from the general sales channel to the airport staff, which can reassign seats as much as they want (or rather, based on various needs), and I don’t think this will be visible online. It can give you a hint, not necessarily a reliable information (it may depend a lot on the airline and its procedures and systems). There could also be waitlisted passengers, gratuitous passengers (staff and their family members who can usually fly at a significant discount, but only if there are seats available). It is usually best to wait until boarding has ended to try to move to a seat.

Also note that if you notice an empty aisle seat, especially with en empty middle seat next to it (or even a full empty row, but those become quite rare), it is quite probable that people in nearby middle seats will be lunging for them as soon as it is obvious they are free.

Also note that at some point between end of boarding and take-off (ideally before they close the doors, but sometimes they run late and do it during taxi), they normally have to count passengers to check that they have the same count as on the manifest. Don’t move while they count. And if they have a discrepancy they will start checking seats which are empty but shouldn’t be and vice-versa. Unless they have other requirements as described above for you to remain in your assigned seat, they can usually handle a few moves, but if they still have count issues they may ask you to move back to make things simpler for them (especially if you moved from one section to another, separated by galleys or the like).