If by "several" you mean "two", yes.
You were not asking about generic systems. But what's the basic contrast to a system with mechanics tied to genre? A system *without* mechanics tied to genre - aka a "generic system". They kind of naturally fall into the discussion.
Not in the least. You can like what you want, it's nothing to me.
Just to be clear - I've said my favorites weren't generic systems. That doesn't mean I think generic systems are necessarily bad. I can have a perfectly good time playing a generic system. But if I have to gauge how much fun the system itself is to work with in play, I just haven't found generics as enjoyable as genre-specific systems. I also think several systems sold as "generic" aren't nearly as generic as they sell themselves to be. And therein lies some of my issues, when you try to cram a not-quite generic system into a genre it does poorly.
GURPS and Supers comes to mind as an example. GURPS basic mechanics and combat rules are not really generic. They're pretty focused on grim'n'gritty (and fairly detailed) combat. This plays poorly with standard comic-book-superhero genre material, in my opinion.
I only quoted this because I'm actually running a GURPS Supers game right now.
I would tend to agree that the game does still have some grit; even when using it for Supers. One of the biggest differences between HERO and GURPS is that HERO starts cinematic and moves toward more realism with options; GURPS starts more ground/plausible/whatever-word-you-want-to-use and then moves toward cinematic with options. Both can move toward the other and feel close, but the underpinnings are still there, and each game has what it tends to lean toward.
That being said, I have been doing pretty well running my game; I just have to run it a little differently. One area where GURPS tends to clash with comic book expectations is that GURPS likes to try to make things make sense; define things. In a comic book, pretty much whatever the writer decides makes sense for the story or the particular character does. For example, Squirrel Girl is one of the most powerful beings in the universe simply because the writers think it is funny, and that's what they write. In GURPS, Squirrel Girl likely gets turned into a red paste after being punched by Hulk.
I think, for me; at least so far, my game is coming across more like a graphic novel with four-color influences. The cliches, tropes, and other things are still there, but with a more serious underlying edge. Personally, I like that. So, for me, I wouldn't exactly say the fit is 'poor' for me; it fits what I want to do. I would say it fits differently though, and, while it's not a poor fit for me, I can certainly understand why it might be called a poor fit by others; in the same manner, I would likely feel HERO system was a 'poor' fit for a grim and gritty game about a Vietnam platoon. In spite of both systems being toolkit systems, they still do have some fundamental differences in their ideology and design. Both can, to some extent, attempt to mimic the other, but both also have strengths and weaknesses.
Though, to continue the discussion, I am someone who believes fluff and mechanics have a relationship which produces the feel of a game. One of the problems I had with D&D 4E was that I actually kinda liked both the mechanics and the fluff; I just didn't feel the fluff and the mechanics did a very good job of working together because (to me) their relationship with each other was often at odds. In contrast, I feel that Edge of The Empire does an excellent job of capturing the narrative and action of Star Wars. In a few cases, I've personally felt that some settings were actually better when using systems that they weren't designed for; an example of that is that I personally feel the whole 'Points of Light' idea and the concept of a world battling against encroaching darkness worked better with a GURPS Dungeon Fantasy game that I ran than it did with the D&D 4E rules.
To answer the OP:
I'd say that Fantasy is most likely my favorite. I'm a big fan of a toolkit system (GURPS) so it's hard for me to clearly answer if the different genres I like use the same system. In many cases they do because I use the same game for a variety of concepts. Though, I do also play other systems, and sometimes other systems are better for a particular concept. In a few cases, I've even mixed a few pieces from different systems; an example of that would be that I've used the concept of Edge of The Empire's light and dark side tokens with a GURPS game.
After fantasy... That's tough to say. I think, in a vacuum, sci-fi would be next, but my personal tastes when it comes to sci-fi don't always match up well with other people, so I typically end up playing something that doesn't suit what I want when I join sci-fi groups. I think sci-fi still ranks so high on my list (being second behind fantasy) solely because I have such a high opinion of Edge of The Empire.
Falling behind Fantasy and Sci-fi would be some kind of modern day campaign; one which is similar to our own world, but has a few unnatural or cinematic elements. Examples would be a war campaign in which the PCs are a group of soldiers who accidentally open a portal to hell, or maybe something based loosely on the video game Bad Dudes; maybe a GTA inspired game where the PCs are an up and coming gang trying to make their way.
After that, I'm open to pretty much anything.