Ask HN: Facebook Web Plugin

I'm starting to see this integrated everywhere https://techcrunch.com/2017/11/07/facebook-introduces-a-messenger-plugin-for-business-websites/ however I can't find anything about the privacy of this new feature. If I visit a site, what stops them from storing my identifying details? I'm aware that sites can track how I've moved through the web via cookies but it's semi-anonymous, this is pushing the limit a bit?