Currently, when you're creating a new template in the Site Editor, you aren't able to pick and choose when template you'd like to clone, instead relying on fallback mechanisms. This was flagged as a current pain point in this post:

I want my two custom taxonomy templates to match the custom post type template. There doesn’t seem to be a way to pick which existing template you’d like to clone when starting a new template. This would be really nice:

1. Select “New Template”
2. Pick Custom Post Type Archive
3. Get to select from existing Archive templates (post type, taxonomy, etc.) as starting points
   As I write this, I realize that I can at least put the body of all three of those template into a Template Part so I only have to edit them in one place.

@jameskoster I could have sworn you had an issue for this exact thing but I cannot find it. It feels related to #41060 and the prior work in #37407