I Have Dual-Booted Linux & Windows 10 On My Laptop And as I mostly game on Linux It's Quite A Headache To Move The .minecraft Folder Around.
Also I Am Sure That It Becomes A Problem When You Setup Minecraft On A New PC / Mac.So, Why Can't Microsoft(Mojang) Add A Sort Of Automatic Saves Folder Backup Which Can Be Switched On And Off? It Would Also Help If Your PC Got Infected With A Virus.The Get The Backup You Could :-
For New PC:-
Download The Launcher At minecraft.net
Log In To Your Account And The Save Folder Should Download.
For Infected PC:-
Go To Your Minecraft Launcher Setting And Click The Restore Past Saves Button
Cloud saves is the term you are looking for I believe. And it would be nice, although I don't see it happening any time soon (maybe the Win10 edition if you have OneDrive or something).
But as a temporary measure you can install to a directory used by other cloud storage like Dropbox, if that works on Linux.
Otherwise, if you have a separate internal drive I don't see any reason you couldn't install Minecraft and point it to the second drive on both OS installs. Possibly. I have very little knowledge of Linux other than basics.