[iOS] Persist Overflow Menu badge data as a pref

The badge data is persisted directly to a new Dict pref.

Bug: 1445203
Change-Id: I19b3012acea33eac77c20946d96ccb3dc5e377e1
Reviewed-on: https://chromium-review.googlesource.com/c/chromium/src/+/4775576


Code-Coverage: findit-for-me@appspot.gserviceaccount.com <findit-for-me@appspot.gserviceaccount.com>
Commit-Queue: Robbie Gibson <rkgibson@google.com>
Reviewed-by: Nicolas MacBeth <nicolasmacbeth@google.com>
Auto-Submit: Robbie Gibson <rkgibson@google.com>
Reviewed-by: Rohit Rao <rohitrao@chromium.org>
Cr-Commit-Position: refs/heads/main@{#1184820}