[Clank][BrowsingDataModel] Retrieve data from the new BDM fetcher

With this CL we pipe the BDM into Clank and retrieve cookies and storage
info from the new BDM fetcher. This is controlled by a feature.

Low-Coverage-Reason: OTHER (Changed signature on many files)

Bug: 254415177
Change-Id: I322ec52f9736cea46236fa8a2cad3ac55086a083
Reviewed-on: https://chromium-review.googlesource.com/c/chromium/src/+/5378371


Auto-Submit: Nicola Tommasi <tommasin@chromium.org>
Commit-Queue: Andrey Zaytsev <andzaytsev@google.com>
Reviewed-by: Christian Dullweber <dullweber@chromium.org>
Reviewed-by: Andrey Zaytsev <andzaytsev@google.com>
Cr-Commit-Position: refs/heads/main@{#1283900}