I have Real ID disabled in account privacy settings and have turned off all Real ID settings in the battle net app. I couldn’t find any in-game settings for Real ID, but this is the only place where it shows. When a match ends and it shows the summary, stats, and talents, when I hover over my name it shows the Real ID.
It only shows for you.
Others can’t see your first+last name in the score tab if you disabled RID in Account Privacy.
(But I agree, it can be a disorienting seeing that)