What happens when I swipe a companion to the right?

They are added to your chosen companions, the list at the top of the Chats tab. Tapping the heart does the same thing. The list marks anyone you have not started a chat with yet, and you can search it and sort it by name or by when you last talked. Swiping someone off the list asks you first, and you can undo it for a few seconds. Nobody else can see the list.