To give another user access to Google Tag Manager you will need to navigate to the Admin > User Management
Within user management you will see who has access to the container, here you can add, remove or manage their user permissions. To add a new user click the plus icon at the type right > Add users
Under the Account Permissions section tick the checkbox next to user then make sure to set the container permissions by clicking Set All.
Make sure all of the boxes are checked and click done.
Lastly, add the users email and click invite.