Access rules are based on roles, rather than individual users.
So a particular role can have particular rights to an entire custom post type, or you can create groups of individual posts to define rights to that group, but, again, based upon the user role.
You can restrict access for individual users with conditional shortcodes, typically so that the currently logged-in user would only see their own content, for example, although you could make it so that only a specific user could see certain content.
Thank you for your answer. I was not very clear in my explanation I think. I want one custom post type with all its posts only accessible by one user (and the admin), and also the new posts that will be made (on the front end with a cred form) assigned automatically to that user (role). Now I have to go into the post and click in the upper right corner which user groups it belongs to, to assign it to that user group (of one user in this case). That is what I would liker to avoid.