I have no idea if players are required to wear Nike gear specifically, but the rulebook does address the issue of visible attire:
“All visible items worn on gameday by players must be issued by the club or the League, or, if from outside sources, must have approval in advance by the League office.”
I’ve never seen a league-issued shirt sans logo. It’s entirely possible that RGIII got special permission to wear this shirt. Or maybe he accidentally put on a league-approved shirt inside out.
Or maybe there's a giant Nike swoosh on his back and we just can’t see it.