Make SearchEngineUtils Profile-keyed.

By default, Search engine choice is linked to Profile - each profile
may use a different default Search Engine to perform searches.

Most of the operations related to SearchEngineUtils are therefore
Profile specific, and while using a single shared static instance
allows us to get most done in Chrome (because Android doesn't support
Profile switching yet), it creates a few other issues, such as:

- SearchEngineUtils cannot track changes of the DSE,
- SearchEngineUtils cannot proactively retrieve DSE favicons,
- SearchEngineUtils cannot report whether currently elected DSE is
  Google.

Currently the most important calls of the SearchEngineUtils require
the caller to supply Profile and TemplateURL, and rather than
notifying the caller about the state change - they require the caller
to poll.

The change:
- removes plumbing of the singleton instance,
- permits multiple clients to share Profile-keyed instance,
- enables the SearchEngineUtils to track TemplateURL changes and
  - proactively retrieve DSE logo,
  - confidently report whether the DSE is Google

Note: SearchEngineUtils continues to use static members internally.
Cleanup of these members is handled in the next change to reduce the
review complexity.

Bug: 1495498
Change-Id: Ib4dd189f15f73e950382564b96e4394acdfa56cd
Reviewed-on: https://chromium-review.googlesource.com/c/chromium/src/+/5035419
Reviewed-by: Ted Choc <tedchoc@chromium.org>
Code-Coverage: findit-for-me@appspot.gserviceaccount.com <findit-for-me@appspot.gserviceaccount.com>
Commit-Queue: Tomasz Wiszkowski <ender@google.com>
Cr-Commit-Position: refs/heads/main@{#1225912}