This is interesting. Not quite on the same level-- but the recent documentary about Archie Manning showed that he would wear little (if any) University of Tennessee attire, especially when they played Ole Miss, while Peyton was at Tenn.
So, yes it is somewhat of a surprise to see Clark in an Ohio shirt. IMHO
I think the major difference here (Archie vs. Clark) is, Ohio isn't in the same conference as Ohio State. So there's no underlying "tension" if you will, for lack of a better word. Then there's that matter of how insane the south takes it's SEC football.
Now if Nick played for Michigan, and Clark had on Wolverines gear, then it would be a HUGE deal IMO lol. But speaking as a father, you saw a man who was proud to show support for his boy.