Plus, it's worth repeating that Firefox sync works without problem across desktop and mobile. The sync [0] and auth [1] servers are open source, and one could choose to use one or both on his premises.
IIRC, browser synchronization was a thing on Firefox well before it landed on chrome.
[0] https://mozilla-services.readthedocs.io/en/latest/howtos/run...
[1] https://mozilla-services.readthedocs.io/en/latest/howtos/run...