No reason a magical catbird hybrid couldn’t be imagined to have teeth too, and no reason a less, er, specialized and bird-y jaw structure couldn’t support them.
 
Intermediate ancestors of birds had teeth, albeit reptilian ones. Modern bird skulls still have mandibles. The bright external beaky structure is keratin - fingernails, hair, that stuff. There are layers with blood vessels in between, and the bill part grows throughout the bird’s life. No reason why the beak couldn’t be drawn serrated, but it’s not tooth stuff.
 
Mammal teeth are all specialized, and kitty teeth have very distinct killing bits and cutting up bits in their adorable little psychopathic faces (ess oo are! You are a little killing machine!). So an artist might map the canines onto pointy bill structures, since they’d move with the jawbone, and keep the carnassials along the jaw for artistic reasons, I suppose. It might even make sense from an evolutionary perspective that some feline fangs might be lost and their job move over to creative bill structures. You know, assuming magical lion/eagle hybrids are going to breed true in the first place.
 
I may be overthinking this.

@Background Pony #12FE
 
@Athaniar
 
@when she felt her wings unfold
 
Hmm..That’s was interesting theory. The keyword is expressions I think. Thanks!

Tsitra draws such insanely cute griffons. <3  
Or maybe it’s just Gabby that’s insanely cute. :p

Maybe they get them from the feline part of their biology?

@Alex Vostox  
It’s a common animation trope, to give beaked characters a wider range of expressions. I’m guessing griffons don’t actually have teeth in-universe.

Yes you. You are cute catbird. But not as rawr as Gilda, sorry.
 
@Alex Vostox  
Since S1 if I remember right. Normally you don’t see them, but the animators probably find them useful.

@Alex Vostox  

 
Technically they shouldn’t but they look more expressive with them.

Do Griffon really got a teeth?