A user can add permissions in this site collection.
However a user can't add permissions in a different site collection. I would like to know why it is not working. The user has full control on both site collection.
Can you please check the group owner! Site Permissions=>Group=>Settings =>Group Owner.
Only Group owners and site collection administrator can add users to Groups
First, go to site settings -> Site Permissions -> Check permissions -> enter the user name and see check whether the given permissions are reflected.