I think if I saw a statue of a cat just chilling, I'd get the vibe. Humans have been entertained by animals doing human-like stuff since we first developed the brains to recognize it.
I know for a fact the name is Dombili. There is no such thing as tombili and I don't know why the wiki page says Tombili is Turkish for fatty.
There is Tombilik which is a small tuna fish and there is Tombik which is similar to Dombili but it's used for cuter things. While Dombili is more like slang in way you would jokingly say fatty.
To sum up there is no such thing as Tombili.
I wish you all a great life.
Well I'm Turkish and used to live in Kadıköy. I never heard anyone calling this cat Tombili. It was always Dombili. It seems someone made a mistake and it stuck with it.