Not sure why wizards wouldn't have distinct identities already. Just going to use 5th level versions as standard for audience who is engaged with development.
Only three classes in the game are oriented to dealing damage through spells.
Most of the spells, possibly all if you spellswap, don't require a to hit roll. They have different typed damage to where Fire Dart and Frost Dart are not the same spell in all cases. They come in different levels so not all attacks are the same.
Druid does not have multitarget spells. Wizard has more than Elf.
There are some buff possibilities.
Druid polymorphs much better, but Elf polymorphs better than Wizard.
Force damage has special rules.
Alertness, Lesser Maze do different things from other effects in game and from each other.
Mage Medallion and Ashenne both open up additional options that will get used.
Meanwhile, get plenty of downside. Saves profile may be different but there are classes with vastly better saves in total. ACs hard to improve, not really worth bothering. HPs low. Spells run out (much less so now).
While tying HP to more options is not the most enthralling thing to me, seems to me that the complaints aren't about identity, they are about damage. If builds averaged 5/10/20 or whatever more damage per round, then what would people have to complain about? (Or, maybe average is a bad way to look at, let's say that could more easily double damage on one round per combat, not double base damage, double damage.)
Terrible defenses? I'd be in favor of more attention given to improving saves as improving AC or HP is a thematic fail. I'd be in favor of additional token slots in something like ring or wrist or charm or whatever, but I have a hard time seeing this fly from a balance standpoint even if it could be thematically justified. For instance, Robe of the Philosopher: +1 Charm, +1 Ring. There's an argument that that's horribly broken since torso slot is pretty much meaningless slot currently to a lot of wizard builds. I had a NM build where I didn't run anything in torso because it was funnier and meant pretty much nothing to the build.
If want to complain about identity, Human Fighter, aka "the easy class". The fighters are sold as hitting more easily, except Fighter only has +1 to hit over Barbarian and far less damage, while DF has no bonus to hit over Barbarian.
They have identities, but I'd say Monk and Ranger have confused ones. Ranger is pulled in two different directions. In some respects that pull to do different things is a good thing. But, melee damage output is just so insane and multiple attacks has a fundamental problem unless you miss that second attack a lot. Meanwhile, Monk identity is supposedly "class with lots of wacky differences" - no weapons, Stun, weird defenses. But, in practice, kill, kill, kill because it can, with things like Stun just making kill even better.