Anime is like fish and liquor, in that periodically people insist I should like it and make me try it again, but I never do. But unlike fish and liquor, or maybe like them, anime seems to lack a commonality that explains why I don't like it. If you consider the breadth of things that are anime, in terms of storytelling style, types of stories, and essentially everything else about them, there's not much that they can be said to have in common apart from that defining trait, the particular animation style. Even that's kind of blurry: there's no particular element that all anime have and nothing that isn't anime has, though there are certainly style elements that are hallmarks of anime (like the big eyes small mouth thing).
But none of those things that define anime are things that I have a problem with. Admittedly, I don't especially like the animation style itself, but there are plenty of animation styles I don't particularly like, but still like specific things done in that style. Animation style is just not a make-or-break element for me.
Otherwise, while there are certainly story elements, themes, images, character types, etc. that are more common in anime, there's nothing you can point to and say that if you don't like that, you won't like anime, because for every anime full of superpowered teenage girls and tentacles, there's another anime that has nothing like that and which wouldn't even be compared to anime if it didn't happen to use that animation style.
Logically, therefore, if the one thing all anime has in common doesn't bother me, and if anime covers such a wide variety of topics and types of stories, then it's inevitable that there must be anime I would like. And yet, every time someone who likes anime and knows me tries to pick some anime I would like, and I give it a try, I never like it. The nearest I've come is to find one that was an interesting story and subject matter, but burdened with a clichéd approach and insipid one-dimensional characters. Still, a few changes and it could have been an acceptable story.
So I wonder if I've just had bad luck with the choices other people have made in an attempt to find some anime I'd like.