I'm not a biologist, nor even a Shrek expert, but Fiona is half-human, so presumably ogres are similar enough to humans to be able to cross-breed and therefore must be mammals. Having said that, Dragon and Donkey manage to breed, so now I'm not so sure...I'm a bear of little brain and easily confused!
I guess it's possible that Shrek is some kind of fantasy, fiction-based thing and ogres, mammalian or otherwise, don't really exist? But in that case, they might as well bf as not! 