How would anything on the internet know about x.x.x.x.y.y…?

Your computer knows it’s connected to an old router because dhcp gave it x.x.x.x address and not x.x.x.x... so it knows it’s running in old v4 mode.

And it can still send outbound to a v4x address that it knows about.

Your local machine isn't on the IPv4 internet if it doesn't have a globally routable IPv4 address.

Your home router that sits on the end of a single IPv4 address would need to know about IPv4x, but in this parallel world you'd buy a router that does.