Second category (taxonomy) metabox for custom post type?

Hello everyone, I’m developing my own theme that uses custom post types. I need to be able to add a second category metabox to that custom post type, so that I am able to select a (single) “main category” (or other hierarchical taxonomy) for a post.

I’m trying to do this as “native” as possible, so if there’s a native way to select one category as main category and automatically assign this post to the category (so that it’s listed when I look for all the categories that the post is assigned to), please tell me how to do that. If there’s no native way: How can I reach what I want as good and clean as possible?

Thanks for your support, guys! ??

Regards,
René