If your game requires a server anyway, then the most platform-independent method is indeed to store savegames server-sided and use accounts with passwords to synchronize savegames between devices owned by the same person. Accessing the savegames on the server could be done via a webservice.
An update for the first week of 2020's:
Cocoonjs was probably a solid solution for this issue in the past but they are shutting down..
a quote from their official web site that says it all
It is with deep regret that we announce that we will stop providing
our beloved service, Cocoon.
We started Ludei/Cocoon because we believed the future of app ...