Thanks @Max_Shop
That was also my first thought. But the only template (in 6.6) that loading „staging-info.html.twig“ is „base.html.twig“
But when looking at the error message, it seems to be looking for the template „video.html.twig“. Okay, I found a include for this file in „box-standard.html.twig“:
{% sw_include '@Storefront/storefront/utilities/video.html.twig'
And comparing this file for both 6.5 and 6.6, the 6.5-version are missing this include. And I had, of course, extended the 6.6 version. 
Well, know I least know where the problem is. 