Due to the way people can change their names freely, I don't see how this could work realistically. Right now as you know you need your ID or to be in-game with someone which makes sense. It's unfortunate but if you've both played a long time, you are likely to come across each other!
Good luck in finding the person you played with!
If you have any ideas regarding improvements that can be done in the game, post them here! Developers read this section frequently.
I still think search based on name should be a thing though. Maybe it won’t help much as people can change their name freely(which I think is a good thing on a game played by children too) but it will definitely help you narrow it down. Also if players had an extra column for a motto after the name, that would definitely help.