WIP: Use GlSearchBoxByType in project selector

What does this MR do?

It introduces GlSearchBoxByType to app/assets/javascripts/vue_shared/components/project_selector/project_selector.vue to align it with the overall gitlab design direction.

See designs here https://gitlab.com/gitlab-org/gitlab-ee/issues/6953 (issue that aims to re-use the project-selector and up-to-date designs)

Screenshots

Before	After

Does this MR meet the acceptance criteria?

Conformity

• Changelog entry for user-facing changes, or community contribution. Check the link for other scenarios.
• Documentation created/updated or follow-up review issue created
• Code review guidelines
• Merge request performance guidelines
• Style guides
• Database guides
• Separation of EE specific content

Performance and Testing

• Review and add/update tests for this feature/bug. Consider all test levels. See the Test Planning Process.
• Tested in all supported browsers

Security

If this MR contains changes to processing or storing of credentials or tokens, authorization and authentication methods and other items described in the security review guidelines:

• Label as security and @ mention @gitlab-com/gl-security/appsec
• The MR includes necessary changes to maintain consistency between UI, API, email, or other methods
• Security reports checked/validated by a reviewer from the AppSec team