Hi John - same requirement on my end. I really like RB's custom tags and was curious if they would sync automatically through cloud sync in the creative plan.
So I tried it on my RB 6 test installation between Macbook Pro (High Sierra), Lenovo Yoga (Windows 10) and iPhone 11 (iOS 13.3.1) and am happy to report that custom tags do sync between these devices.
That is, new tags created, assigned or changed on Lenovo are propagated to Macbook and vice versa. All tags are correctly displayed in iOS RB and can be assigned there (via "info" button on main screen), it is however not possible to create new custom tags in iOS. Syncing to iOS and Lenove was quite fast (~1 min max), on the macbook it took a while longer and I even restarted. Not sure if the latter was even necessary, in the end RB on Macbook was perfectly in sync with the other devices.
Hope this helps...