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 stay
As others have said, this is a deliberate limitation of popup UI.
Instead, you could inject some HTML into the page which loads the content you want in your popup into an element which hovers over the existing page. You will have to implement the close functionality yourself, but it will persist.
Have a look at e.g. how keyframes.app
has done it: https://github.com/mitchas/Keyframes.app/blob/master/Keyframes.app%20(Extension)/src/inject/ui.js