This sounds very limiting to me. I typically have several instances of ArcMap running simultaneously dragging layers back & forth between them. Maybe once I get better versed in Pro there are other, better options for this, but right now this feels like it will be an issue.
No, there aren't really better options, we are stuck with workarounds until it is addressed. This concern was raised very early on by beta testers. It is a known issue they want to resolve and are working on resolving, it just isn't resolved yet. I can't say what the odds are of having it addressed by final release, but I am not holding my breath.
Not to nitpick on semantics, but they are important in this case. A user can run more than one instance of ArcGIS Pro, just not on the same machine at the same time. The current per-user licensing model allows a user to run ArcGIS Pro on up to three machines, unless offline licensing is enabled, and then the limit is one.