Magic Fang is a bad example since it doesn't specify that the target must be an animal (and in fact it's often cast on monks). Animal Growth will fail on your magical beast for not being an animal, though. Doomman47 is correct that they'd need to include text like that in Celestial Servant for it to count as an animal for you.
I doubt the author of Beast Speaker thought about it and decided not to put in such text, especially since the feat requires Handle Animal; more likely it didn't occur to them that it was needed. These things happen. If you're in a home game, I would talk to the GM about inserting the relevant text from Celestial Servant into Beast Speaker.
To be fair, as long as you don't lose or trade Share Spells away, it's mostly fine regardless.
It's part of what it does : allow you to cast spells on your AC even if it normally wouldn't affect creatures of its type. Be it Enlarge Person or Animal Growth.
The very similar Monstrous Mount is written somewhat the same way, for that matter, as are pretty much all the feats and abilities allowing non-animal "Animal" Companions. The ones including more forgiving wording are rare.
Not that many friendly spells target animals specifically, and you usually (if not always) can get around that thanks to Share Spells.
It can definitely be a problem for things other than spells though, for sure.
But magic is usually what matters most to us.
It is something to keep in mind when looking at companion archetypes.
You can use this skill on a creature with an Intelligence score of 1 or 2 that is not an animal, but the DC of any such check increases by 5. Such creatures have the same limit on tricks known as animals do.
So, handle animal can work on non-animals as long as they don't have a high intelligence.