It is a player guild.

Linkshells used to serve this purpose as best they could, but they lacked many features a true guild has in many other games. Linkshells were always just chat channels, and only served as guilds because nothing else existed. Going forward, linkshells will be left as chat only because they no longer need to do anything else, Free Companies having taken all the guild functions.

You can join as many linkshells as you want, but can only join one FC/guild at a time.

All the talk of involving Grand Companies is just a way of explaining through lore that any player can join any guild, regardless of their city-state affiliations.