I had the same issue with a normal Pi 5 Install. I had to manually create an auth key in my Tailscale admin console and then use the command line in HA to set it. I used the SSH add-on (don’t forget to set username and pw in config) to get command line access.
To create the auth key, in your Tailscale admin console, go to Settings > Personal > Keys and create one, and copy it to clipboard and maybe drop in a file so you don’t lose it as once you finish in Tailscale admin you can’t get it again.
(Should look like: tskey-auth-kc9pvdq79F11CHTRL-LNT8WzHsKA575RQeuNGyA5gL4tDYiPVqZ)
I had to delete the add-on completely then reinstall, and BEFORE starting…
Go to the add-on page, open the config tab and set to log mode to debug. Don’t start it yet.
Get your container information
watch -n 2 ‘docker ps -a | grep tailscale’
(Should look like “a0s5d924_tailscale”)
With your container code and the auth token from your Tailscale admin console run
curl -X POST -H “Authorization: Bearer $SUPERVISOR_TOKEN” -H “Content-Type: application/json” -d ‘{“options”: {“auth_key”: “TAILSCALEAUTHKEYHERE”, “log_level”: “debug”}}’ http://supervisor/addons/YOURCONTAINERCODEHERE_tailscale/options
Then start the add-on
ha addons start YOURCONTAINERCODEHERE_tailscale
Check logs
ha addons logs YOURCONTAINERCODEHERE_tailscale
Look in logs for https://login.tailscale.com/a/YOURTAILSCALEAUTHKEY (See @Oleksandr.C image above)
Click on it right from the console, login to your Tailscale admin console and approve Home Assistant as described by @Oleksandr.C
After which you should not be presented with a Login prompt (that fails) in the add-on webui, but should again see that it’s running and connected to your Tailnet. Good luck!