Support syncing between browsers / machines

AFAICS, Tampermonkey only supports script syncing, not storage syncing. I'd vastly prefer to avoid reimplementing some storage syncing layer if we can. Maybe [Violentmonkey](https://violentmonkey.github.io/faq/#why-are-third-party-sync-services-used-instead-of-native-ones) supports this natively? I haven't checked yet.

Also ideally any solution should preserve user privacy, i.e. not disclose unencrypted mappings over the network to third parties like cloud service providers.

Start by reviewing the userscript's current storage behavior and investigate whether Tampermonkey or Violentmonkey provides native storage syncing. Define a solution that synchronizes storage across browsers or machines without exposing unencrypted mappings to third-party cloud services; the issue is complete when that approach is agreed and implemented.

Written by the indexing model from the issue text.