Problems embeding and linking videos in a SharePoint 2013 site

Hi JakobRohde. For SharePoint, there are two ways you can embed videos from Stream into SharePoint - and that depends  on the version of SharePoint.  You can use either the embed code directly or use the Microsoft Stream webpart directly (if you are using SharePoint Online). 

Can you confirm if you are using SharePoint online or on-premises?  It is possible that something is missing in the on-premises cases which would help us to figure out.

Addiitonally, it would help if you can share the embed code you are pasting into SharePoint. It's possible that the version of SharePoint you are using is blocking the fullscreen. Can you also open up the browser console (by hitting F12) then selecting the fullscreen button when the console is open to see if there are any errors.