One WordPress posts, two URL’s

Does anybody have advice how to build this?

I have a WordPress-site (let’s say xyz.com) with a Divi-theme. Everything works perfect. On the website it shows all post-messages.

Now I want another WordPress-site (let’s say abc.com) with also a Dive-theme but some elements should be differently styled. For example the menu is blue on the first site and I want it green on the second site. Of course a lot more styling then that. Moreover the second website abc.com should only show post-messages if a message has a certain category, let’s say ‘ShowOnABC.com’. The first site (xyz.com) shows all messages, the second only that category. Also some pages are different on both URL’s.

In the ‘ideal world’ I would log in to WP-admin of xyz.com and make a new post message and set the category ‘ShowOnABC.com’ if I want to show it on both sites. If I don’t checkmark that category it only shows on xyz.com

Does anybody have a good idea to handle this? And is it possible?