To get a tag partner, you have to gain their trust. You can do this two ways. One way is to duel one of the main characters, and the other way is to give them sandwiches (that they like).
After you do one of these actions, you should hear a little ringing sound. If you don't, skip a day and try again as needed. You can get the ringing sound to happen about six times a day on a good day. The reason you're doing all of this is to fill up the little hearts next to the character profile in the database. There are eight hearts, and six ringing sounds fill up one of them. When you fill up seven or eight, the character should ask to be your tag partner.