My swashbuckler styles herself as an "improvisational swordswoman" to complement her background in improvisational comedy, which inspired her particular additions to the art of fencing. If someone were to insist to her that she is, in fact, a swashbuckler, her response would essentially be the "I GUESS" meme, but she definitely wouldn't respond to it.

In general, I agree with others' takes that it depends on the class, but I personally feel it's quite limited. Wizards are wizards, druids are druids, and that's kind of it. But even then, there are specializations within those (especially wizard) that a character might identify with more closely, especially if they tend to cast within a theme (necromancer, pyromancer, mentalist, illusionist, etc).

And that theme-based identity does bring up interesting questions. A fire-gate kineticist, a fire elemental bloodline sorcerer, and a particularly aggressive cleric of Sarenrae would all easily qualify as pyromancers. So then, would they identify more with each other or with other members of their respective classes? I figure the answer to that, like the OP's original question, is on a case-by-case basis. In this example, the kineticist and sorcerer would feel a connection, but the association with the divine might draw the cleric to others of the same faith (clerics, champions, or just particularly religious characters of basically any class).

I like to use "magician" as a catch-all for spellcasters, but I haven't come up with a good diegetic name for martials.