My T usually tells me general chit chat things as well. She did share once that she had a similar trauma issue and startles too, but didn't get any more detailed. I think it's okay to have T share; I think it made her a little more 'human' for me (i tend to have her on this pedestal)
But I agree, if it does bother you that he shares with you, just let him know; since it seems like you have a good rapport, it sounds like he would understand