Is there any particular authentication your users use? (SAML / Tableau Auth)
In relation to the popup, from my own experience with embedded views, the popup does close so if this is not happening you may want to create a support case so we can investigate.
However in terms of permissions the popup will still only give them a login to Tableau Online as if they went to https://online.tableau.com through the browser so they will not get any extra permissions. They will only be able to access what is allowed by their permissions.
Thanks David! The issue solved on its own. That same afternoon the pop-up began working as it should again.
We still don't know what caused it, or whether the problem was on our end or not. Should it happen again we will create the support case.