A class has never been a PC's identity to me and honestly I'm not sure what kind of identity you would want. Sometimes a blank slate is the best option. On the other hand, there are plenty of fighter subclasses, many of which (to me) have far more flavor than, say, the wizard subclasses. A cavalier feels different from an arcane archer from a rune knight in ways that the school of illusion vs enchantment never will. Strength based fighter wielding a maul or dex based throwing daggers, dual wielding or using a shield to help protect your fellow PCs? Archer or melee? The list of variation goes on.
Wizards in the other hand? School of evocation? Cool, you get some nice benefits but you can still cast that charm person just like every other wizard. That's boring to me, at least at the class level (bladesinging is the exception, but I've never seen it). The choices they make will vary, but they'll always be intelligence based and have access to the same list of spells. If they take a feat, the vast majority of times it will be warcaster.
Which doesn't mean I have a problem with either the fighter or wizard. They're not perfect because nothing is, I just disagree that there's not a narrative difference. A fighter in plate trying to chop you in half with a greataxe feels a lot different than that guy wearing studded leather trying to make you into a pincushion with their arrows.