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What services are you talking about? The history Android permission, or something else? Certainly I don't think there's any such API for Chrome.

As a Firefox user, I thought of Firefox Sync first, which seems to have enough documentation to make a third-party client: https://developer.mozilla.org/en-US/docs/Firefox_Sync

I find it a bit surprising that Chrome doesn't have something similar (or is the relevant sync code just too buried in Chromium?).