Algolia search results page: handle docs multi-instance + contextual search

[slorber]

🐛 Bug Report

https://v2.docusaurus.io/search/?q=doc

The Algolia search results page has currently a few problems:

• It does not include the next/unreleased version in the dropdown
• it does not support docs multi-instance (for example ios v1+v2, android v1+v2+v3 on same site)
• it does not use the new contextual search feature (see feat(v2): contextual search, dynamic Algolia facetFilters #3550)

I consider this bug low-priority, as:

• it keeps working fine for many sites without multi-instance (or with only one instance being versioned, like D2 website)
• this page is likely not used often
• might not be trivial to solve work, for little ROI

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Notes for when we'll fix this:

• We should make the SearchPage in Algolia being decoupled from the theme, so that it can be used with new themes too, that won't be based on Infima. (Algolia theme would provide the search behavior, but the "parent" theme will provide the styling of the page)

• We should remove the legacy useVersioning() hook in favor of the hooks provided by useDocs.ts, as we can retrieve more info (like version labels) with it.