Who can delete a branch in VSTS?

If you are both Project Administrators, you should check if the user has been set permission separately.

In Version control Tab https://account.visualstudio.com/project/_admin/_versioncontrol?, check if the user has been set permission separately:

• Check in the repo level: select the repo (such as Git2 in below example) -> check if the user has been set permission separately -> if yes -> check if Force Push as Deny -> if yes -> Clear explicit permissions.

  

• Check in your two branches separately: select each branch -> check if the user has been set permission separately -> if yes -> check if Force Push as Deny -> if yes -> Clear explicit permissions.

  

Check Force push (rewrite history, delete branches and tags) permission for repo under Settings > Version Control is set to Allow. Force push to a branch, which can rewrite history and this permission is also required to delete a branch.