It's not cookies. It can't be used to track you across sites the way that a cookie can, as all the embedding site needs to do is ping the site the cookie is for, and it's done. How do multiple sites access the same localStorage?
With an iframe. Although Safari already has protections in place against this scenario: https://webkit.org/tracking-prevention/#partitioned-third-pa...
I can't try myself right now. But last time I checked it's not at all that easy. What am I missing?