Is there an adjective for "bear" in a similar sense as how "feline" and "canine" are adjectives for cat and dog?
As @mari-lou-a said (in a now deleted comment), according to dictionary.com it is "ursine"
of or relating to a bear or bears.
An example usage of "ursine":
we noted that a preponderance of the evidence supports an ursine origin for the giant panda (Giant Pandas: Biology and Conservation)
Another synonym might be "bearlike". Wiktionary has one definition that says:
Like that of a bear