Allow syncing user data across multiple devices

Today the seed phrase is the only information we work from when restoring a user's account on a new device, but we could persist some settings and configuration in a way that improves experience.

In particular, the sync strategy should work between desktop (extension) and our beta mobile clients, differentiating this issue from #4561, #2547, #1459, and #6318.

This issue is not opinionated on the exact sync strategy, and instead includes a research task of choosing our best approach.

A previously developed comparison table is here:
https://docs.google.com/spreadsheets/u/2/d/1X_fvJmPBoxI5riPFcAn3nJ1oE7RXvKnEKsohxFjmJ30/edit?usp=drive_web&ouid=113766561670897768939

Start with the comparison table linked in the issue and review issues #4561, #2547, #1459, and #6318 to understand existing approaches and constraints. Research a sync strategy that supports the desktop extension and beta mobile clients, then document the selected approach and its trade-offs as the definition of done.

Written by the indexing model from the issue text.