Is it possible to create a custom user permission request dialog for HTML5 APIs (e.g geolocation or getUserMedia) so they look the same across all browsers?
Thankfully, no!
If you were able to create your own, you might phrase the question (and buttons) misleadingly. Just like you can't style the onbeforeunload popup.
Anything that is outside your sandbox, and you're trying to get access to, has to be controlled by the browser UI. Otherwise it'll just be complete and utter mayhem.