This PR makes it so we hide the plan badge next to the username in the
user menu when there are organizations.

We instead show the plans next to each organization:

<img width="257" height="317" alt="Screenshot 2026-04-13 at 1 29 26 PM"
src="https://github.com/user-attachments/assets/0111e7ff-af09-48ba-a043-9e25c2dd24bd"
/>

Closes CLO-649.

Release Notes:

- N/A