I avoid any "remember me" options anywhere because it allows sites to track you even when not logged on. As to how it works, your devices have what is known as a mac (media access control) address. It is a hardware address unique to every internet device out there. The IP address is assigned to your device dynamically but the router you connect to translates data from this address to your mac address to actually communicate to you. Marriott likely records your mac address permanently when you select "single click" which means that if you ever connect to them even if you don't log in, they know it is your device. Another feature to steal your privacy.
Of course, what I generally do is use a mini-router (about 2"x 2" x 1") which connects to them for hotel stays so they pick up that device's mac address (temporarily as I don't select "one click"). All my devices then connect to my mini-router which is the only thing that sees the mac addresses of my devices. Then each device doesn't have to set up a wi-fi connection over and over. Moreover, my mini-router uses VPN (virtual private network) to encrypt traffic so Marriott and its partners don't see what sites I connect to.
I was a computer and network security and privacy person before retiring so I don't like private companies, who have no limitations, invading my privacy although it is impossible to stop much of it without being a troglodyte.