Organizations are any game-supported group of characters.
Different games have different names for these. Most fantasy-styled MMO(RP)Gs use "Guild". Some like "Clan". I (Jae) prefer "organization", since it's pretty neutral.
A character should be able to be member of multiple organizations. When you think of "Guild" and "Clan"... it's obvious how someone could be both in a guild and in a (distinct) clan. The clan being his extended family. While the guild being, for example, the thieves guild. Or the assassins guild.
In a different context, a character might be an employee of a corporation, let's call it "BigCorp Inc." and be a member of a certain department of said corporation, let's say the "Department 9" (counterintelligence ;) ). Which would mean her being a member of the organization "BigCorp Inc." and the suborganization "Department 9".
Yep, organizations could have specific relationships to other organizations.
Another example: there's a "Police Force" organization which requires new members to be of a specific minimum level (without "levels": new members must meet certain minimum abilities/skills requirements). And there's a second organization, the "Police Academy", where everyone can "enroll". The leaders/officers of this second organization would of course be members of the "Police Force". After some time, a member of the Academy either leaves, or leaves and joins the Police Force.
Limits on multiple memberships: Depending on game/world/setting, multiple memberships would/could be limited. It generally would not be possible to be an employee of two different companies. Or to be in several guilds at once. A character could be restricted to only one guild membership, or specific guild memberships could be mutually exclusive (if you could be a thief and an assassin, of course, you would (likely) be able to be in both guilds. But the assassins guild and the healers guild would likely exclude each other (even when you could learn healing skills as a primary assassin, the healers would be unlikely to accept you)).