It's Complicated.
Foxes are definitely Japanese natives, as are tanuki, even if China also had the idea of monster foxes. On the other hand, domestic cats were brought over from China to Japan around 500 AD, to help protect Buddhist scripts from being eaten by mice, and legends of monster cats came with them. Some might have been brought over earlier, and there was another big import of them later (1100ish), but at the very least normal domestic cats are not native Japanese.
To further complicate things there are two cats that are native to Japan, apparently holdouts from the last ice age, the Iriomote cat and Tsushima cat - but they never expanded beyond those islands and mainland Japan didn't really know about them. Iriomote's in Okinawa which is a separate (blended) culture, and Tsushima's a remote island half way to Korea. So anyhow, cats aren't considered Real Traditional Japanese, even if they're cute and fluffy and people love 'em.
So now I was curious as to what medium to large animals are 'real' Japanese, which in practice means anything there from late Yayaoi period (1000 BC) or earlier and besides those two cats, foxes, and tanuki, we've got bears, deer, hares, macaques, serow, wild boars, badgers, martens, stoats, and weasels. Lots of mustelids! There were two wolves (Japanese wolf and Hokkaido wolf), but they're extinct. And too many birds and marine animals to bother listing, plus those migrate.
So it's kind of like ramen which is technically Chinese, but the Japanese have changed it so much you could argue it's Japanese now even if some Japanese still call it Chinese food... Zanma could totally decide that 1500 years ago is plenty old for bakeneko to be Japanese enough, but on the other hand this might be why they haven't shown up at all.