I haven’t had to do any of this before, so this is all just an idea of how I’d approach something like this.
This should be somehwat straightforward providing that verifying a video ID in a URL is enough. If you need visual evidence that the video is correct, well, good luck.
The player likely has some sort of JavaScript API, right? Can you use it to query the current state? You might be able to run the appropriate JS to query its state direct from your tests. You may even be able to hook into various events the player supports to add classes or attributes to the container for the player that would be easier to assert against, though with anything you’re writing yourself you might ask, “How do I test THAT?”
Again, I’d look to see if the player has a JavaScript API that could help.