If I open my extension popup then I open another window or tab following the popup does not stay open if I return to it.
Is there a way to force it so the popup stays open?
As a user, you currently cannot force the the popup to stay open. That is a UI decision the UI team made. If you want to want to force a setup, you can have other way to show this by changing the popup icon, open a new tab when it requests, or new popup view for registration.
As a developer, inspect the popup, and it will stay open.