Did the laws restricting hunting rights to members of nobility extend to all creatures, or just a select few like deer? Because in Westeros inns often seem to serve boars, but never venison.

Good question!

It’s a bit of both. Historically speaking, there were forest laws setting aside certain regions as game reserves (so restricting hunting rights for all animals in a given areas) but also royal “licenses to empark” which specifically gave noblemen the right to enclose a bit of forest and kill a certain number of deer in a given period.