Search: Allow sync manager queue to account for multisite context

[rebeccahum]

Description

Currently, multisite context switching in the sync manager queue is not supported right now. This adds it per Automattic/ElasticPress#204. Should have no real material effect since get_sync_queue() replaces $this->sync_queue and has the same return.

Changelog Description

Fixed

• Enterprise Search: Allow syncmanager to support multisite context for queues

Pre-review checklist

Please make sure the items below have been covered before requesting a review:

• This change works and has been tested locally or in Codespaces (or has an appropriate fallback).
• This change works and has been tested on a sandbox.
• This change has relevant unit tests (if applicable).
• This change uses a rollout method to ease with deployment (if applicable - especially for large scale actions that require writes).
• This change has relevant documentation additions / updates (if applicable).
• I've created a changelog description that aligns with the provided examples.

Pre-deploy checklist

• VIP staff: Ensure any alerts added/updated conform to internal standards (see internal documentation).

Steps to Test

[sonarqubecloud]

Quality Gate passed

Issues
0 New issues
0 Accepted issues

Measures
0 Security Hotspots
0.0% Coverage on New Code
0.0% Duplication on New Code

See analysis details on SonarCloud

[codecov]

Codecov Report

All modified and coverable lines are covered by tests ✅

Project coverage is 30.43%. Comparing base (188bfd0) to head (f39df35).
Report is 26 commits behind head on develop.

Additional details and impacted files

@@              Coverage Diff              @@
##             develop    #5969      +/-   ##
=============================================
+ Coverage      30.41%   30.43%   +0.01%     
  Complexity      4812     4812              
=============================================
  Files            288      288              
  Lines          21132    21120      -12     
=============================================
  Hits            6427     6427              
+ Misses         14705    14693      -12     

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