Terms now get properly created and assigned.
The queries seem to be working correctly.

On the PHP front this is OK, now working on the JS front.
Since this was imcomplete (I previously missed all the meta.theme references in the template-part JS files) I converted this PR to a draft so we can keep working on it.

If this is using a taxonomy now, can we use the automated handling we'd get for querying and response inclusion by marking it as show_in_rest?

If this is using a taxonomy now, can we use the automated handling we'd get for querying and response inclusion by marking it as show_in_rest?

Thank you @TimothyBJacobs! I added show_in_rest to the taxonomy now, but I'm afraid I don't quite understand what you mean by "use the automated handling we'd get for querying and response inclusion", could you please elaborate?
Right now I'm exposing wp_theme in the JSON response using this: https://github.com/WordPress/gutenberg/pull/24720/files#diff-51f3ab0134282ae12e1994208a3ab6feR261, and then I use that instead of the meta we previously had. So templatePart.meta.theme just becomes templatePart.wp_theme. Are you suggesting there's a better way?

Now that the taxonomy shows in REST, you should be able to query for it automatically using the rest_base of the taxonomy.

It should also now be automatically included in the response as the rest_base too.

@TimothyBJacobs
Ah OK, I see what you mean now.
The problem with the default responses however is that they use the term-ID, and what we have here is the term-slug.
So I don't think we'd be able to get what we need without an additional REST call... Instead of doing that, I simply added the term-slug in the response.
Note: after your comment I realized that there was a conflict with default responses. By default it was returning wp_theme: [ 2 ] so I renamed the slug to wp_theme_slug and it doesn't overwrite the wp_theme anymore in responses.

If you feel this can be done better then please feel free to push commits here, you are far more familiar with the REST API than I am!

Superseded by #27016
Closing this one since the other PR is up to date with the latest structural changes and it will be easier to work on.