You can't _really_ categorize things. You can mostly sort of do it, but it never actually works out, because categories are human stuff, and human stuff is subjective and thus different depending on who you ask. Sometimes unclear even to the same person.
I can say dogs have five legs and the only reason most people will think I'm wrong is that society has come to categorize limbs based on function instead of length.
limb (n.2) late 14c., "edge of a quadrant or other instrument," from Latin limbus "ornamental border, hem, fringe, edge," a word of uncertain origin.
Just thought of the limbic system and how that might be related, doesn't seem to be about function but about positioning against a core system
But you're right, limbs are not defined by function.
> If you call a tail a leg, a dog still has 4 legs.
I can't think of a useful definition of legs that includes the tail. I can for calling a recording of a musical piece a song.
Dog is a categorical abstraction over individuals. The properties of a category tend to have a complex relationship to the members of the category. This is true even if it is indisputable who the members are and what properties the members have.