It should work:
https://www.esri.com/arcgis-blog/products/arcgis-online/sharing-collaboration/answering-your-questio...
If the partner organization is using a different authentication system, does this affect the collaboration?
No. With partnered collaboration, the ArcGIS Online sharing model controls access to content regardless of using built-in authentication or SAML logins.
Which makes me think that there may be an issue with the way the apps are setup.
Have you embedded content in these apps? e.g. a website of https://www.shortkey.maps.arcgis.com/dashboards' in a story map? It will attempt to sign into shortkey.maps.arcgis .com, when the url should be maps.arcgis.com or the item that they can access via the group.
You might have better luck searching for this issue as described for public web apps, as it's essentially the same issue. E.g. the below post is for public content, and they identified the issue based on web traffic:
https://community.esri.com/t5/arcgis-online-questions/public-web-app-requires-login/td-p/1064983
Esri support will be able to help explore the issue within your environment i.e. over screenshare or if you provide a login. Difficult for community if it's not public.