We can name our pets, but we can't name our mounts, which we spend a great deal of time on if they're permanent. Poor unappreciated nameless mounts!!! :(
So that everyone can tell the difference between pet names and mount names perhaps the mount names could be in two parts as a wizard's last name is? The first half of the last name could be words like thunder, night, fluffy, dancing, reaper, etc. while the last half of the mount name would be body parts like hooves, claws, feet, fins, tail.