Welcome to the catch-22 that dragonborn fans have been forced to endure for 15 years now. Our preferences don't have enough traction or historical representation to deserve inclusion, so they never get a chance to develop a foothold, meaning they can't get any traction or historical representation.
The key difference, at least for me, is that gnomes DO have traction within the D&D space. That traction just isn't very interesting for most people, for several reasons. Whose fault that is, I've no idea (well, mostly I think it's the fault of Dragonlance and World of Warcraft, but that's a separate topic.) But the fact of the matter is, even though there are other interpretations of gnomes, the only one that's made any headway in the D&D-and-its-children space is "eccentric expert and usually comic relief/hyper-cutesy twee." That's why they're disliked so much; people who would want to do something serious with them are just usually choosing to play something else.
And I get it. I get that that sucks if you're a gnome fan, for exactly the reason I gave at the start. I've been there, with people (even theoretically friendly/positive ones) straight-up telling me I should be glad that dragonborn got included in the PHB at all.
But the thread was asking why this response occurs. I gave what I consider to be the full and complete answer: they're short (short races will always be less popular than human-height or taller races, except for their potential as twee or comic-relief options), they're often portrayed even by players who love them as completely non-serious, usually comic relief or zAnY wAcKy HiJiNkS types; their major, public-facing appearances in well-known fiction almost exclusively support the latter, with World of Warcraft being the prime offender (since it's much bigger than D&D has ever been, having had easily tens of millions of players over the many years it's been a thing.)
That doesn't mean you shouldn't play them. Hell, it doesn't mean you shouldn't campaign for gnomes to get a different take, something more interesting than the crappy typecasting they've gotten. You should totally do both of those things. But those three reasons are why most players won't play them or even outright dislike them, and it's very hard for individual folks (like you and me) to change any of those things.