This is Dallas from Shopify's Social Care Team.
When a video is embedded in your store it's hosted by Vimeo or Youtube. So it's brought over to the Shopify store it's still technically a youtube video that's just been pointed to a website. For that reason, Youtube is the one who designs the layout of the video and Shopify, unfortunately, doesn't have any access to change that feature.
My suggestion would be to reach out to the Youtube team to see if there was a way to get that removed, but considering every blog/website of every level around the internet with a youtube video attached has that layout I can't say that there is a way to remove it.
I completely understand how frustrating this is because I too tried to remove that from my website with no luck. If you have any further questions please let me know!