How to make custom posts to behave like taxonomy?

I would like to have a database of fashion models on my blog.

The models section will be located under /models/. There shall be a loop of recent posts with models as well as the full list of models names.

Singe page of a model shall have URL: /models/kate-moss.

Each model’s single page should be a custom wordpress post, including basic description, photos and special fields like age and birth date, weight and height etc.

On the single page template below the description of the model there shall be a list of posts related to the person.

One more important thing: when an author publishes a post they should be able to simply choose the related person from the list with a checkbox.

In other words the person is neither custom taxonomy nor custom post, it is both. But how to implement it?

Have a taxonomy, with terms that are the same as the custom post type. Then in your taxonomy template, grab the custom post type that has the same post name as the current term slug.

You can automate this slightly by having the celebrity term automatically created when you add the celebrity post etc

Refer to this answer for how to do that:

How to add automatically keyword to taxonomies when a post published, and assign them to the post