You need to finish at least one of the 4 regional problems main quest items to gain a sage avater. Once this happens the godess statues will offer you the choice of "Enhancing your bond" with them. Like the normal light blessings you need 4 sage's will to do one. I haven't chosen to use the 5 I've found so I've not seen what that actually means. Maybe quicker refresh times on using their powers...not sure.I did Rito first so I will gain one other before I decide which to upgrade first.
Note the Master Sword requires 2 wheels of staminia so if you should try and unlock the statue if you didn't invest correctly. I was got tier lucky and went in blind not knowing what the requirement was. I happen to have exactly enough on first try and didn't have to hunt down the light dragon again and her first spawn location was near a tower so made getting to her very easy on first attempt.
I ended up completing this before I had started any of the Sage related quests. The Master Sword has no displayed damage stat but feels like pre Master Trials sword in terms of damage output and how often it has to recharge. Yes you can fuse stuff to it and it's interesting how it handles that.