OK, so after further investigation - in short: Organisations which you are not a member of are not shown.
In my situation, the organisation is that which the owner of the project (the person who set it up), belongs to. Presumably, because I am an admin but not a member of the organisation, I do not have the permissions to see this.
Rather than tell me this, the UI goes to the extent of actually telling me incorrect information - that the project has no organisation. This a valid, but entirely different state!
Extremely confusing and I would argue - a bug. Many aspects of the UI and processes refer to an organisation, which always says "No organisation" for myself, despite the fact that on the same screens for the owner, it clearly states the assigned organisation.
The only way I could figure this out was to share screenshots of the same screens between the owner and I.
In regards to the oauth user type, I have to assume that it applies to the owner's organisation.