Hi k5v9nmc,
Thanks for coming to Apple Support Communities for help today with Fitness+ and using it with your Apple Watch and Apple TV. It sounds like it won't pair for you but will for your wife.
Please see: Set up Fitness+ on Apple TV
"Choose who is working out
If you use Family Sharing, you can easily switch between family members in the Fitness app on Apple TV. Friends or family outside of your Family Sharing group who are Apple Fitness+ subscribers can also work out using your Apple TV.
- Open the Fitness app on your Apple TV.
- Select your name, or select Other if you don’t see your name.
- If no one is signed in on your Apple TV, you may need to select Sign In in the Fitness app first.
- To switch to another family member or guest, select the account icon in the top left corner of the Fitness app, select Sign out, then select another user."
Have a fantastic day!