Use relative URL for projects to avoid storing domains

What does this MR do?

Uses project_path helper to get project path while logging a project for frequent access as this removes any dependency on domain root which can change leading to broken links.

Why was this MR needed?

Older implementation of Projects dropdown stored full project path (including domain name), and this could be a problem when domain is changed or HTTPS is turned without auto-redirect leading to invalid links stored in localStorage.

Does this MR meet the acceptance criteria?

• Changelog entry added, if necessary
• [ ] Documentation created/updated
• [ ] API support added
• [ ] Tests added for this feature/bug
• Review
  • [ ] Has been reviewed by UX
  • Has been reviewed by Frontend
  • [ ] Has been reviewed by Backend
  • [ ] Has been reviewed by Database
• Conform by the merge request performance guides
• Conform by the style guides
• Squashed related commits together
• [ ] Internationalization required/considered

What are the relevant issue numbers?

Closes #40895 (closed)