1e magic-users could be a little unique, since they didn't pick spells, but got some randomly at 1st, then just made the best of what they could find - a party with multiple MUs could always share spells and lose said distinctiveness, of course.The thing is I don't think it's actually lost much since about 1985 and both Unearthed Arcana giving you specialists with chosen spells and predefined stories meaning not every wizard was different. It just hasn't gained much since then.
2e specialists got to choose a new spell with each level, but they still had an opposition school, so they were each different.
Then, opposition schools went away, so even specialists gravitate to the few best spells, regardless of school.
For all that players chafed at memorization, weapon/armor proscriptions, difficult requirements for casting and so forth, they at least were ways, however arbitrary-seeming and heavy-handed, to make classes, and thus characters, different.
3e experimented with having only consequences instead of proscriptions, with ways to work around those consequences, and, really, should have been considered a failed experiment...