With the block template system in edit-site, WordPress will have the ability to handle both theme templates and user templates contiguously. There should be a way to identify when a template has local modifications compared to the vanilla template.
Template selector, no modifications:

Template selector, "home" has been modified by the user / site:

Showing a tooltip:

This could also be the basis for doing template reverts (restore a theme default) as well as "theme switches".
Implemented:

Nice one @epiqueras !
Is _Edited_ a better term than _Customized_?
"Edited" could be confused with "Unsaved Changes".
@epiqueras do we have anything left here or can we close?
We can close.
Most helpful comment
Implemented: