Almost every fighter today is a bit multi-role. Even the MiG-31 can toss a Kinzhal at you.
As for the F-35, it was designed to a requirement that was described contemporaneously, by the program boss, as 70% A2G and 30% A2A. For instance, the customer didn't ask for supercruise, for an all-aspect IR AAM to be carried internally, or for superior air combat maneuverability to the F-16 or F/A-18.
There's also the question of how the aircraft is to be used. If, for instance, the Marines are to attempt an amphibious landing against an adversary with a couple of squadrons of Sukhois, they're going to go in with a CVN and its air wing. Six fighters in an MEU are simply not enough to do serious A2A and any A2G. It's been explained here (or in the carrier thread) that the RN plans to do fleet air defense with the T45. I don't hear much from the US Navy about the wonders of the F-35C in A2A.
This is nothing unusual: most operators of the F-15E and later variants task and train for A2G.