Each server is independent, but they are connected together into a single network.
It's like with telephones: you use your phone provider's SIM card to make a call, even if the call is to someone on another phone provider, because it's your provider that lets you get onto the network.
Or email: if you have an account on Gmail and you want to send an email to a Yahoo address, you wouldn't try to sign in on Yahoo. You would sign in on Gmail and send the email to a Yahoo address.