Check headers for current organization when accessing v4 api

What does this MR do and why?

This MR extends header-based organization selection to the v4 API, matching existing functionality in other contexts.
GitLab users belong to multiple organizations and need to perform actions across different organizations during dogfooding, and this ability will be required for instance admins for managing organizations.

References

Related #548319 (closed)
Related #576243 (closed)

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