If you want to keep both names, you actually need 3 tags.
Steps:
1. Pick one of the names and change that pet's name to something random.
2. Tag the name you just removed to the pet you want it on
3. Tag the name of the second pet to the first pet
If you don't care about one of the names:
1. Tag the name you want to keep away from the pet it's on and name that pet whatever you like
2. Tag the name you want to keep onto the pet you want it on
The second method means you'll only need 2 tags, but you'll only keep 1 of the names, so it's your choice really.