Draft: Port left side of project header to Vue

What does this MR do and why?

The right side of the project header has been fully ported to Vue, however, the left side is all still HAML. This ports the rest of the header to Vue as well.

MR acceptance checklist

Please evaluate this MR against the MR acceptance checklist. It helps you analyze changes to reduce risks in quality, performance, reliability, security, and maintainability.

Screenshots or screen recordings

Screenshots are required for UI changes, and strongly recommended for all other merge requests.

Before
After	WIP

How to set up and validate locally

Numbered steps to set up and validate the change are strongly suggested.

Closes: #457848