The same outfit was most noticeable in season one, right before Ned Stark was executed on Joffrey's orders.
The costume change for Jon was a clear indicator he was allying with the Stark name and turning away from his life at Castle Black. This symbolism carried all the way through to the Battle of the Bastards and Jon's coronation as king in the North. He may not have the name, but the North recognized him as a Stark.