That is a really good question, but unfortunately I can’t answer it 100%. Quota is really being applied for any API interaction, however it is not applied to Iframe API when you embed videos (because you are not providing any keys or tokens). If you open Google Developers Console and navigate to API section of any project, there will be Enabled APIs tab. Inside you will see quota per API. Therefore interacting with YT Analytics in not the same as interacting with YT Data API. These quotas are counted per project.
What I do not know is how to set up those per user and seems like from this console there is no way to do that (moreover it seems that there is no way to do it at all, see below). I believe you can search for such topics in Google groups or contact them directly. I have heard somewhere (though I am really not sure if it was related to Google APIs) that if you need much more API calls for a project, then you may contact Google team and discuss this personally. For example, here is some info on usage limits for Google maps https://developers.google.com/maps/documentation/javascript/usage It clearly states that if your app generates too many traffic, Google team will contact you to discuss payment options. Some more info on billing: https://developers.google.com/console/help/new/#billing Money makes the world go around, as they say.
Hopefully that helps and thank you for the feedback!