Yes, really.
Providing a list of synonyms doesn't make calling oneself a fighter any less relevant than calling themselves a soldier or any other term.
Telling people how to think about their characters is bordering on a badwrongfun speech. I would have no problems with a player saying in character that they're a fighter, just as I wouldn't have a problem with them saying they're a soldier or a mercenary.
It's ridiculous to assert that one term is wrong just because there are numerous other names for it. Telling a person they can't say, "I'm a rogue," when they're playing a rogue character is just as silly and nonsensical as saying they can't call themselves a fighter if they're playing a bard.
Nor is either any more or any less logical in the context of in-character discussion than any other self-referential term that a player decides to use for their character.