Hey @ingo,
We’re sorry for the confusion, the background video is not centered on mobile and that’s not a default as there’s no option to position and most mobile browsers do not allow background videos to play and it’s not good for mobile page load speed.
That is the reason why there is a Poster option. Please set a poster image to replace the video on mobile. It’s best that you use a portrait-oriented image to match the aspect ratio of a phone screen.

Another thing about background videos is they should not contain important information. Background videos are for decoration only.
If you want to show a video on mobile, create a new Section and add a Video element to it. Only show the section on mobile screens using the Hide During Breakpoints feature. Similarly, hide the Section with the video background on mobile using the Hide During Breakpoints feature.
Hope that helps.