Yes, to offer that functionality it would need a cookie. No, the website doesn't need to correlate that login and all activity related to it to everything else you ever did on the internet to build up a profile about you. It is technically feasible (even easier) to offer the first ("remember me on this one site") without doing the second ("track me across all sites").
Well, technically it's all the same, it's a request to FB servers with an user id and token. Decoupling that relies entirely on Facebook's internal policy. Since they are a public company now, not using that dsta is leaving money on the table.
