Yes, because the main similarity is the fact that even though both characters exhibited morally questionable and reprehensible actions and behaviours, they became fan favourite characters because of how entertaining they were (and, yes, in large part due to male horniness centred on an attractive, fictional female character).
Harley's popularity led to her character and backstory being greatly expanded upon to give greater understanding of why she became so infatuated with "Mistah J" and turned to a life of crime, and in the last 10 or so years she's undergone something of a reformation by breaking away from him and becoming not just more of an anti-hero character, but also one of DC's most popular characters.
We don't yet know what Quinn's reasons are for doing what she does and being how she is, but I see it playing out similarly to Harley in that there is a compelling reason for all of this, however messed it might be, and she has it in her to turn things around and become a better person, yet still retain that edgy demeanour.
I'm not getting what your point is, then. If you're saying Quinn is a lot like Harley before she went mainstream but the audience was generally more forgiving of Harley... doesn't that imply there must be something different about them beyond just the longer name?
As I said, Quinn is always going to have a harder time getting audience sympathy because her transgressions aren't overshadowed by the Clown Prince of Crime. Harley being a sidekick meant her terrible actions frequently looked tame in comparison. Quinn isn't a sidekick, so her actions are forced to stand on their own.
Plus, Quinn is arguably the main antagonist in the game thus far, and her actions are frequently much worse than the competition. Quinn easily manages to out-asshole Tommy, IMHO. Tybalt is her only real match, but they almost never interact and the one time they did Quinn still came off as the bigger jerk because of the whole "ditching Hell Week to peddle drugs" thing.
Beyond that, though, the other problem is that Quinn is not nearly as memorable a villain as Harley was. Harley had an instantly iconic visual aesthetic backed by an inspired voice actor (who could in turn play off Kevin Conroy and Mark Hamill); Quinn has a pretty standard goth look and is silent. As a girl, Harley immediately stood out from Batman's mostly male cast of characters; Quinn is in a game dominated by female characters. Harley got to piggyback off one of the most iconic villains in comic books; Quinn is a solo act doing fairly typical "mean girl" things. It's not really a fair match for Quinn.
Again, I'm not trying to say that Quinn is a worse person than Harley, or even that Quinn is a bad villain. I just think it would take more than just a sympathetic backstory to turn Quinn into Harley.
Of course, I've always been skeptical of Harley as an anti-hero, too. So perhaps I'm not the demographic to worry about.