[Refactor] Make ExtensionErrorController a keyed service.

To facilitate the ongoing refactoring of ExtensionService and the
separation of extension management logic, ExtensionErrorController is
being transitioned to a browser context keyed service. This change
decouples it from ExtensionService, enabling independent use by various
extension classes. This is the initial step in porting
ExtensionErrorController to desktop Android, with subsequent UI code
porting to follow.

This cl is pure refactoring effort without changing extensions
functionalities.

Bug: 401047740
Change-Id: I87bb69d0d3bd61b4015f4138ce5728d492395722
Reviewed-on: https://chromium-review.googlesource.com/c/chromium/src/+/6331508
Reviewed-by: Stefan Kuhne <skuhne@chromium.org>
Reviewed-by: Devlin Cronin <rdevlin.cronin@chromium.org>
Commit-Queue: Jenny Zhang <jennyz@chromium.org>
Cr-Commit-Position: refs/heads/main@{#1429835}