I added a user under "Access" tab on report customization but they still can't overwrite the report, they can only save as. Am I missing something? They have full permissions for Report Customization under their role

This sounds like a question of Ownership, not Role/Permissions. I don't believe another user can overwrite someone else's Report. If they were the owner, they could overwrite with a "Save", but as the owner is likely someone else (you?) they are only given the "Save As" option. As an option, you could change the Owner of that report to another select user, which should allow them (and only that user) to overwrite the report.

@Dan Curtis It was a role/permissons! You can overwrite someone else's Report but seems as if I was missing the Granting access to reports permission from that specific role. Now they can save and overwrite. There's an audience tab and an access tab on custom reports. Audience can view and save as but they can not overwrite. Access can overwrite

Aha! Glad you found it!