An organization is a form of grouping of resources and accounts (users). Resources created for the particular organization is only possible to view and modify by the members of that organization with the exception of system administrators.
Users of an organization are accounts added to that organization. Accounts can be added to multiple organizations.
The user selects organization when logging in. All actions are then based on/applied to the selected organization. It is however possible to log in to multiple organizations at the same time but with different access tokens. One access token cannot be used for multiple organizations; a token is bound to the selected organization during login.
There is a concept of organization administrator, which is only a regular user with full permissions for all resources within that organization. Multiple users can have full permissions to an organization.
The Varnish Controller has no theoretical limit on how many organizations can be created. And the same applies to how many organizations a user is part of.
See authorization examples for examples of organization management.