Is there a fundamental difference in how one would approach tanning the hide of this animal vs. tanning the hide of that animal? Or is the process generally the same across all species (for terrestrial mammals, that is)?
Not really. More differences are about what you want the leather to be – soft ladies gloves, shoe uppers, shoe bottoms, chariot wheels (!), and the inherent toughness differences in each animal’s skin under their fur.
Briefly what’s wanted is more or less cross-linking of the collagen. Tanning (leather) - Wikipedia Making it more or less stiff and more or less decomposition resistant.
Professional trad tanner here. The OP’s question is such a sweeping one it is hard to tackle, but: there are massive differences between the skins of animals, even within species due to age, sex, season, locale etc. Also true that there are underlying principles governing the process of turning various collagen matrixes into various leathers.
I think Tanya Hyde would be a great username for someone.
PM sent to the OP.