Subpages for Custom Post Type. Does that even exist?

When you register post type with register_post_type, you can set hierarchical argument.

hierarchical(boolean) (optional) Whether the post type is hierarchical (e.g. page).

Allows Parent to be specified. The ‘supports’ parameter should contain
‘page-attributes’ to show the parent select box on the editor page.

Default: false

Note: this parameter was intended for Pages. Be careful when choosing it for your custom post type – if you are planning to have very many
entries (say – over 2-3 thousand), you will run into load time issues.
With this parameter set to true WordPress will fetch all IDs of that
particular post type on each administration page load for your post
type. Servers with limited memory resources may also be challenged by
this parameter being set to true.

As you can see, its default value is false, so the CPT is not hierarchical. Set it to true, and your CPT will be hierarchical.