IMO the mark of a good game (or good game concept) is that it inspires you to think about it and it's mechanics. I feel like with what you have here already I can see a great game. Some ideas I've had while playing:
-Have pawns only be able to move in straight lines and attack diagonally, similar to real chess but not limited by direction. Right now, pawns are very often more dangerous than anything short of a queen, which seems wrong.
-Have some sort of limit on the number of traits a unit can have, and make it possible to cancel or remove gained traits. It seems way too easy to just stack every good trait on one queen and use that to win every game. Also, it'd make specializing/balancing units a more important consideration.
-Units with shared traits get a weaker version, e.g. an unit in a relationship with one that has "sleepy" would only get +1 health; which makes the initial trait a unit has more important and influences how the play builds relationships.
-Maybe have separate "light" and "dark" square bishops? Not sure how this would balance out, but it would be interesting.
-Definitely do... something with kings, to make them more important. Probably not like in actual chess, since that'd get a little silly, but maybe have them give some type of buff to the rest of the team while they're alive, or heal units next to them, idk.