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Change Profile to BrowserContext/BrowserContextHandle.

Note that SiteEngagementServiceTest has Chrome dependencies
and can't move to //components.

Bug: 1147274
Change-Id: Icaf2a007bc427d5f5276189c9161169a63f51f86
Reviewed-on: https://chromium-review.googlesource.com/c/chromium/src/+/2601454
Commit-Queue: Evan Stade <estade@chromium.org>
Reviewed-by: David Trainor <dtrainor@chromium.org>
Cr-Commit-Position: refs/heads/master@{#840847}