[extensions] Remove uses of extension_api_unittest.h (17/30)

This CL is part of a batch of CLs migrating all uses of
extension_api_unittest.h to more recent alternatives.

This change corrects the file under the path:
   /chrome/browser/extensions/api/identity

This CL was uploaded by git cl split.

R=alexilin@chromium.org

Bug: 394840
Change-Id: Ic20e950539775b610ff00adbe1f97ca3be5250a9
Reviewed-on: https://chromium-review.googlesource.com/c/chromium/src/+/4382187
Reviewed-by: Alex Ilin <alexilin@chromium.org>
Commit-Queue: Alex Ilin <alexilin@chromium.org>
Auto-Submit: Claudio DeSouza <cdesouza@igalia.com>
Cr-Commit-Position: refs/heads/main@{#1124078}