Firefox

Yes, this is absolutely just a possibility for a website to do it. Actually it's probably also quite complicated technically, but there are multiple services for recording precise user behaviour including all mouse movements on a website, so I would imagine there's something for this, too.

What are the specific mechanisms of action we're concerned about?

I was thinking about the website's code running some light checksum on all the resources it has downloaded and loaded into the browser, and if it differs then upload the diff. I think it should work to find groups of people with a similar browser setup, but maybe it would fine just as browser fingerprinting too.