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After install the ARC app, update the equivalent Chrome app to add the
badge on the icon. When the ARC app is removed, update the equivalent
Chrome app to remove the badge on the icon.

This is the follow up for crrev.com/c/1833006 to implement the the
change in extension_apps.

This CL breaks unit tests when testing multiple profiles:
NoteTakingHelperTest.AddProfileWithPlayStoreEnabled
WebAppInstallTaskTest.InstallWebAppWithParams_GuestProfile
The reason is IsArcAllowedForProfile returns true for any profile under
tests. crbug.com/1010137 is used to fix that. Since that is
out of the scope of this CL, for this CL, a work around is implementted
to look up |user_to_profile_for_testing_| map before going to the
default primary user.

Bug: b/141388525

Change-Id: Ic11965d72ec6ad1636cd0765f57cadc377bd5479
Reviewed-on: https://chromium-review.googlesource.com/c/chromium/src/+/1824307
Commit-Queue: Nancy Wang <nancylingwang@chromium.org>
Reviewed-by: Nigel Tao <nigeltao@chromium.org>
Reviewed-by: Dominick Ng <dominickn@chromium.org>
Reviewed-by: Xiyuan Xia <xiyuan@chromium.org>
Cr-Commit-Position: refs/heads/master@{#704093}