There’s no cookie left identifying them as a particular user, most likely. I don’t know enough about your wesbite and how you handle user authentication to say for certain though.
This is generalized because I don’t know enough about your setup, but: when they log in initially your website stores this in the session cache on the server. So long as that client remains actively connected, your server will maintain a copy of the session locally. Generally, this is done with a cookie that is placed on the user machine that has some unique identifier in it that is sent along with every user request.
I don’t know the exact time it happens, but when you close the browser, the browser will go through the cookies it has stored and clear out any that are expired or don’t meet any conditions requiring them to be kept–some people configure their browsers to dump all on close. Without this cookie to transmit on the next request to your website, your server will see that request as a new, unique user.
I am not aware of any Patreon provided code that would handle generating a cookie on your server.