Some extra details from Adobe Support:
I have tested this several ways including directly in the desktop browser and it always loads the video with the "Auto" setting for quality. I'm not sure how that is selected by the youtube service (depending on browser/device or whatever parameters) but it always brings up maximum resolution on desktop and 360p on mobile no matter what I tried.
So this is not really related to AEM Mobile but to how youtube feeds the video. Unless the video is already at the desired resolution, the viewer will just display whatever it is fed.
I'll ask around for a workaround but I do not have high hopes I'm afraid.
Solution: Use Vimeo iframe embeds instead of YouTube. They display the correct resolution without any extra parameters. Took a little extra work to get them uploaded, but it was definitely worth it.