If you want to be a furry, buy yourself a Hat of Disguise and activate your fursuit whenever you want too. Pick whatever race and class you want. The rest is anxiety about powergaming.
You may not be aware of it, but there are two things in that statement that are offensive. First you group all furries into a narrow stereotype ( fursuiters ), and then dismiss their concerns as frivolous and unimportant. I don't blame you, there are very few furries actually talking to people about themselves. Most reflexively cringe every time the topic is brought up outside furry circles, no news is good news most of the time.
There's actually very little you can say about furries as a whole. Basically just that we all like the idea of human/animal hybrids to some degree. Some see their furry forms ( called a fursona ) as their real selves, others see a fursona as a symbol, and others don't even have a fursona. Some fursona's are very deferent from the real person, others aren't deferent at all. Some have sexual feelings about furries, others don't. Some think they're reincarnations of animal spirits. Some would welcome the ability to actually become their fursona, others only think of it as a fantasy. Fursuiters are little deferent than cosplayers. They're a vibrant part of the furry community, but just that -
a part. Some that put on a fursuit feel a spiritual connection, others just see it as a costume.
As to what being a furry means to me, it's complicated. I see tigers as a strong symbol of my perfect self. Strong, curious, independent, fearsome, and beautiful. There is also a bit of a notion that this is the form I'd chose for myself, birth made me one thing but this is what I'd chose. I don't think about it too often as changing your species is a
long way off, but I would like to see us get there. So there is a bit of transhumanism in there too. I also am not a costume person, I don't see the point. They don't look real and they are very uncomfortable. I seek out games that let me carry out this fantasy. When I'm playing one, like the Elder Scrolls, I feel comfortable and happy. It welcomes me in and lets me feel like the real me. So having a cat race in D&D is a big reason why I'd play in the first place, and being cat-like is the whole point. So reskinning another race wouldn't work.
D&D is a combat heavy game by default, so survivability is an important consideration. No one wants to play a game where you feel like you're being handicapped. Lessening the impact of ability scores is the easiest way to do that, which is why it's being brought up. It's not my ideal though, that would be having every ability score equally contributing to your survivability, but in a deferent way. That would make any combination of ability scores viable, even an even spread between all of them. It's just that it would require a real heavy rethinking of ability scores in D&D to make that happen, and I don't think there is a will to do that.
Also, while I'm talking about being welcoming to furries, it would really help if there was a canine race. The most plentiful types of furries are canines and felines, so having those two represented would be great. Having an open ended animal/person race that let you pick from a list of features to make any type of animal person would be ideal, but just a canine and feline race is a good start.