I just created a new 1.16 world and I got a texture issue with horses. I was using a resource pack when creating the world, but when I turned it off the issue remained. I tried reloading chunks and saving & quitting then joining the world again, and the texture issue remained. It looks like horses are using the fox texture.

I spawned in horses with an egg and they had normal textures, so it only affects horses that were generated naturally.

I also tried breeding these horses, and their child also had the same issue.

Hope this is helpful and the issue is resolved!