In regard to Elise from Sonic 06, more than anything else, the key problem with her is that the game asks a lot of the audience in investing in her character and never really gets around to effectively justifying that.
There's really nothing inherently awful about her, but there's also little really make her a standout character, which is a problem when a third of the game is oriented around her relationship with Sonic. Her design is something you'd see in a generic JRPG, she's not really that deep emotionally, and she lacks so little agency that rescuing her makes up the bulk of Sonic's plot. The novelty of her as a non-Eggman human having a dynamic with Sonic is about all she has going for her and even that only amounts to so much given how pat the interactions between the two end up being.
I think a lot of the backlash against Elise is one way or another connected to this. Now the design clash of her vs the rest of the cast didn't help, and most of the Western audience at the time was always going to laugh and/or cringe at their faux flirtation given the differing regional opinions on human/anthropomorphic animal romance. However, I think those problems were very exacerbated by the aforementioned failure of the game to earn its emotional moments with her. For many fans, if they weren't going to be engaged by what she does in the narrative (and the gameplay itself not being nearly strong enough at least distract from the dubious story) the only thing left to do is snicker at the visual silliness of her kissing Sonic and lament that the title spends so much time on her.
I'm reminded a bit of Adam Malkovich from Metroid Other M, another character that a game asks the audience to really take in and respect while utterly failing to make him interesting, likable, or remotely worthy to the deference Samus gives him. In both cases, the failure to effectively realize the concept of what they were going for amounts to audiences either rolling their eyes /laughing at with Elise or just outright despising their existence with Adam. In her case, it's no surprise she ended up symbolizing 06 more than the likes of Silver (lucky him) because her as the bold new idea for the Sonic series that completely fails due to botched execution sums up that game so perfectly.