Unfortunately this is a limitation of the GitHub oAuth API and how it deals with permission, or scopes.
In order to get a list of your Organisations, we need to request user permission scope. This scope includes asking for write access to your profile, including SSH keys.
We never write anything to your profile. Hopefully in the future GitHub will make adjustments to their oAuth permissions to allow us to request read-only access that allows us to get a list of your organisations.
We've got a longer article on the subject here.