When I open some of my Pro Projects containing feature services from ArcGIS, Pro does not prompt me to login to my Portal. Instead, the data connections are simply broken. So unless I remember to login to my Portal connection before opening my project, I have to manually fix connections to all my layers from AGOL. ArcGIS Pro remembers which portals I've added, but not the login information. Here are some details that may be relevant:
This doesn't happen for every project. Normally, if a project contains data from a portal connection, Pro will prompt me for my login information if I'm not logged in already.
All help/advice is welcome, thanks in advance!
Our setup is the same with the exception of an organization URL for our AGOL account and using a virtual machine for Pro.
On your portals tab you can sign in (and use the checkmark to have it remember you) to ArcGIS online but then make your portal URL the active connection. After signing into your portal URL you should be to use the checkmark to have it remember you as well. This workflow has prevented me from experiencing your situation.
Hope that helps. If it doesn't, maybe someone else will have a tip.