It is possible to create as many relationships to the User type as you want, each with is own name (and meaning).
Unfortunately, a basic user relationship doesn’t have the same functionality as the Assignment extension (e.g. notification, permissions etc).
You can read some related discussions here
Also, there’s an extended disussion about users and their functionality here
In general this is our design mistake, we should add Person field instead of Assignments extension (and I hope we’ll do it in the future). So far @Chr1sG suggested the only workaround with relations to User Type.