Well it’s an interesting idea. People already are animals just as giraffes or elephants, we’re just a different species - but still we are part of the Animal kingdom. So in a way, people ARE furry from birth, in having intrinsic animal-like qualities.

I think the draw is a lot deeper especially when contrasted against people who don't consider themselves furry. The idea that a person "doesn't feel like they had a choice in the matter of being furry" is actually very common, and I can vouch for that feeling as well. Luckily I don't mind that part of me. I wonder how people that reject it feel about themselves - I suspect a lot of the loudest anti-furry people fall into this category, similar to the self-hating LGBTQ+ right-wing.

It’s odd that most furry costumes basically look alike - I mean, basically with big eyes and ears and big mouths

This is mostly a matter of practicality and logistics from my understanding. Even with giant eyes it's difficult to see out of them, and the big mouths probably need to be matched to the eye size so it doesn't look bizarre. The full head itself also needs to be big enough to fit over a regular human head. I don't think everyone intentionally wants the costumes to look like they do, but there's probably a handful of factors preventing more realistic costumes from being common, like cost increases for custom designs. I'm not really into the fursuiting part (many aren't), so this is just secondhand info.

It looks like fun - except aren’t the costumes kind of hot after awhile?

From my understanding, they get very very hot. I think the main reason people get them is to have fun goofing off at cons or doing photo op stuff. They cost upwards of 3-4k IIRC and they're handmade, so it's probably something you have to wear carefully.