Define "Identity" because I don't think that's very often realized in champs. Sometimes yes.... Sion is a good example. His kit and playstyle make you feel huge and powerful. Zed for all the QQ set at him FEELS like a shadow ninja. Fast. Bursty. Mobile. Often grossly unfair. If he wanted to fight fair, he wouldn't be a freakin' ninja... and a BAD GUY at that.
But then look at someone like Aatrox. What the holy hell IS Aatrox? His lore makes him into a mytical war god-type thing that inspires men to fury and strength they never knew they had... but drives them mad. So he... jumps around and auto attacks, with a revive passive and looks like Marilyn Manson with a shower curtain on his back. Nothing about his kit works.... either with his lore or his identity. I think this is a flaw in design... Aatrox feels rushed, generic and bland.
On the other hand, sometimes the result is from player input. Take the champ Aatrox WISHES he was... Tryndamere. He's a big, scary, angry barbarian who rages and refuses to die until he rips you in half. Ok. I can see how his kit is meant to invoke that. He has a heal that he needs to be doing damage to build up... he is all about auto attacks... he has a gap-closer to get in close and a slow/debuff to try and force you to fight him. Except that Tryndmere is often used as a glorified splitpusher who avoids fights for most of the game to beat up helpless minions and towers. Was he designed that way? Not really. But players often USE him this way, even though it's not really fitting an aggressive barbarian who wants to fight to say "you guys fight mid, I'll just push top and run if they come up here.". His kit is used for a playstyle not fitting his seeming identity because players decided it was a better use of him.
Then there's the dreadful Cass rework, where Riot decided that her "core identity" was "lategame hypercarry" and not "cool snake mage that poisons people". Instead of trying to emphasize her theme as a sly, poisonous person both reveling in and disgusted by her transformation at the same time... they instead focused on her entirely in mechanical terms. This is why a lot of people hated the rework. Because they liked Cassiopiea's old identity and new cass has a different one. Now she's an in-your-face lane bully who spams her E and her other two spells are only there as an afterthough to LET her spam her E. Her poison doesn't matter anymore and she now has a gimmick more appropriate for a powerhouse aggressive mage like Xerath or Ryze... someone who comes into lane and says "I AM MAGIC, BITCH! RESPECT IT!". She's not a sly, poisonous manipulator anymore... at least not gameplay wise.
Annoying as he is, TEEMO has a very strong identity and he's one of the few unique champs out there. He's built as a scout, stalker and hunter. He can't yolo at you. He has to plant traps... think ahead... hide and ambush... and basically... cheat. He's an underhanded little dude, but he'd HAVE to be. He's 3'1" and his weapon is a blowdart. Teemo is the only champ left with long-duration traps that actually matter, and that's his identity. To do well as Teemo you need to know where your opponents will go and lay in wait... traps in place... ready to jump them. Teemo without his shrooms is a squishy little yordle who dies REAL fast. But he'll play mindgames with you all day. Can you really get him, or is he baiting you into a minefield? Did he really leave lane or is he stealthed nearby waiting? Teemo DOES feel like the splitpush sort... putting the mission objectives above the glory of fighting and getting kills. He evades you... he lures you into bad spots and he puts you down with as little risk to himself as he can. His identity and kit mesh perfectly.
Some champs have a strong identity that their gameplay reinforces... some are at odds with their own archetype, and some could change drastically and still fit their "identity". Kass is a guy who saw the void, was changed forever by it, and now he's fighting back against it. Ok. Nothing in that screams "I can teleport everywhere". None of the void creatures teleport around. Malzahar doesn't either, and he's supposedly the void's main guy! So why does Kass? He's hugely resistant to magic damage. Ok, well that's nice I guess. Half the void champs are AD sorts, so his passive makes him better at killing Lux than most of the void monster... Not something that needs to be kept to keep his identity. He DOES have that bitchin sword though... and that's pretty iconic. Ok, so let's make Kassadin into a flash-stepping melee type who voidwalks in and rips you apart. That seems to be how most people see him. How about letting him keep his powerful mobility ult, but making him far squishier with few or NO spells that can be used at a range. He can have his silence back... but now his Q is melee range. Give him more damage, less durability and less range and give his ult a cooldown of about 10 seconds or so.... BUT have it cooldown in 2 seconds or so if Kass gets a kill. Ok, so now he's a bursty, mobile assassin who needs to make sure he'll take you out when he all-ins or he's stuck in there. If he plays it smart though he can pop in and out like he used to. This is all just theoretical, but it's just an example. You can keep a character's identity and change how he functions as a champion.