[Subscriptions] Move the in-memory cache to the storage

It is easier to manage and update the in-memory cache in the
SubscriptionsStorage instead of in SubscriptionsManager because
we don't care which operation leads to the cache change, instead,
we just align the cache with storage layer all the time.

Bug: b:266822296
Change-Id: I99a6d4e9794305ea3d623507650ebf978e9895a4
Reviewed-on: https://chromium-review.googlesource.com/c/chromium/src/+/4317514


Reviewed-by: Matthew Jones <mdjones@chromium.org>
Commit-Queue: Zhiyuan Cai <zhiyuancai@chromium.org>
Cr-Commit-Position: refs/heads/main@{#1118881}