Why is chicken considered meat for the purposes of keeping kosher but fish isn’t?

I was having a discussion with a sikh friend about why my parents will eat fish but not meat today, and I shrugged and said that it’s probably because Christianity is a knockoff of Judaism, and Judaism doesn’t consider fish meat. Then he said that islam considers fish halal, but I countered that, again, it’s one of the kids that cheated on the homework.

This, unfortunately, got me thinking, and that never ends well. If the reason you’re not supposed to eat dairy on chicken is that it might look like you’re eating it on a different meat, as I’ve heard, then why doesn’t that apply to fish? If you bread and deep-fry something, you can’t tell what’s inside. How come you can make a kosher fillet o fish but not a chicken sandwich with cheese? And if it’s about looking like you did something you’re not supposed to, why do they make kosher turkey bacon? Non-dairy cheese on a kosher burger or real cheese on a vegetarian burger? I’ve seen that before, so why can’t you put cheese on chicken when you can put it on fish? Based on the reasoning I’ve heard, it doesn’t make any sense at all to me.