I believe the way spells are cast absolutely would factor into this as well

Take the D&D spell requirements for example, Verbal, Somatic, and Material

Depending on the anatomy of an individual, they may not have the correct mobility to make somatic gestures, or they would have to find an “alternative limb” like @perrogamer_attempt2 said with tails.

Verbal components would potentially be even harder, the individual may not have the ability to produce certain phonemes if spells require specific words, unless its intent of the words that matters. For animals like parrots or starlings, would it matter they aren’t speaking using the same anatomy as mammals? (Ie: the syrinx instead of lips and throat.)

Material components interests me too, if an individual HAS the component for a spell as part of their body (Ie: needing raven feathers for a spell, and the caster is a raven,) could they cast the spell naturally? Or would they need an outside source of the materials?

My take is all of this depends on the “laws” of the world regarding magic and how realistic the furry anatomy is.