How to add a link to the file manager in a user's profile?

How to call the file manager from an iframe by providing it with the user_id?

You can use a Media Field (Joomla Media Field or JCE Media Field) to provide a File Browser in the user's profile, but you cannot do this directly with an iframe.

and does each user have a specific directory?

You can use variables in the File Directory Path to restrict a user to their own directory. Please see - https://www.joomlacontenteditor.net/support/tutorials/editor/setting-the-file-directory-path