Regis was super excited, embarrassingly so. Think awestruck young dad had Regis actually be able to BE a normal awestruck young dad. He was also very proud because the thing that led to Regis eventually putting 2 and 2 together was Noctis rescuing him after he pushed himself into stasis and unconsciousness trying to use barrier/Wall magic to protect the town they were in from attackers. Regis waking up to Noctis having fallen asleep at his bedside was also a bit of a tip off, though Regis, fool that he is sometimes, blew it off as a tired citizen being concerned for his king.

I suppose he wasn't too far off there. Noctis IS a citizen of Lucis and he WAS tired from using the dregs of his own magic energy to stabilize his father's magic levels. LOL!