Let’s see now. Cats are felis and dogs are canis. The -ine suffix denotes “of” or “related to” so “feline” and “canine” would mean “of felis” or “of canis”
The Africal elephant is Loxodonta and would be “loxodontine” and the Asian elephant is Elephas and would be “elephine.”
However both are of the family Elephantidae so could be “elephantine.”
“Pachyderm” is not an official classification (being polyphyletic - that is, being little more than a hodge-podge of not-terribly-related organisms). Rhinos, for example, are more closely related to horses than to elephants.
However, having said that, yes, rhinos, elephants, and hippos are all informally called “pachyderms” (because of their thick skin).