Would like the ability to group contacts without creating new chat rooms

The Amazon Chime app does not have any way to group contacts. (You can put them in "favorites" and then re-order them how you would like. But, the groups would not be labeled.)

Besides using a chat room, the only other idea I could think of for you would be to start a multi-party chat with the group of users. (Admittedly, that's only helpful if you want to chat with them all at the same time and it's not much different than a chat room.)