The idea is that a user can post as many horses as he/ she wants and can post as many searches as he /she wants.
The user's search post types are public visible.
If he/she wants he/she can "apply" a horse (one of his existing horses or a new) to a search of another user (user 2). So in the post relationship form it should be possible to ADD a new horse or select one of the user's existing horse post types.
That user 2 (owner of the search post type) will then see the relationship and be able to SEE the horse post type of the other user (user 1).
"So in the post relationship form it should be possible to ADD a new horse or select one of the user's existing horse post types."
It is only possible to add existing posts to a relationship using a relationship form. If you want to give the user the option to create a new post and add it to the relationship then they will need to do this through a regular post form.
The Relationship form is only for setting existing posts in a relationship. Essentially to achieve your goal you will need to use 2 forms.
Ok thank you (I thought it was possible as you can do it in backend).
So but then, how can I connect an existing horse with a search post type in a frontend?
as an extra ; Can. I make a form that automatically connects a horse with the 'current' search (if you add the form in the search post type singel)
I make a form that automatically connects a horse with the 'current' search (if you add the form in the search post type singel)
No you won't be able to automatically connect the posts based on the search. What you can do is to click to the page that you want to connect and add the relationship form to that page's template and it should allow you to connect that specific page to any horse that you are searching for in the form.