feat(organizations): add badge to profile dropdown TASK-979

[duvld]

🗒️ Checklist

1. run linter locally
2. update all related docs (API, README, inline, etc.), if any
3. draft PR with a title <type>(<scope>)<!>: <title> TASK-1234
4. tag PR: at least frontend or backend unless it's global
5. fill in the template below and delete template comments
6. review thyself: read the diff and repro the preview as written
7. open PR & confirm that CI passes
8. request reviewers, if needed
9. delete this section before merging

📣 Summary

Add the organization badge to the profile dropdown.

📖 Description

Tiny reshuffle of elements to make it match the designs.

👀 Preview steps

1. Have an MMO with some users in it
2. Ensure the feature flag mmosEnabled is true
3. While logged in with a user in an MMO, clicking the profile icon should show the organization badge (in blue) under the user's email

[notion-workspace]

Add org name badge to app header nav

[jamesrkiger]

Since the only thing we are changing based on where this component is being used is color, let's just have the component take 'badgeColor' as a prop. That way if someone ends up using a light blue org badge elsewhere in the app they won't need to make a new class for it.