If I buy a game that has multiple platform support do you have to buy each version or can you install on multiple platforms?

You get all of them in one go! :D
But.. do you want to install a game on multiple systems? O_o

Well, it only feels strange to me to install 2 copies of the same game on different OSes.
(if it's only for himself)

If you duel boot it would be a great convenience, if the save file is stored in a logical place and not "hidden".

You can do the same thing on Steam. Kinda awesome, if you ask me 8).

On all digital storefronts for PC today, you buy the game and access to all its versions. If GOG offers a Linux and Windows version of a game, you'll get both. If it offers a Mac and Windows version of the game, you get both. If it offers Windows, Linux and Mac versions, you'll get all of those.

Before purchasing ANY game, check the "Works on" section on the right to see if it works on your preferred system.

One more note. If you're a Linux user (or someone who can access Wine), you can download the Windows, Mac, and Linux versions to your desktop if you bought the game from GOG. If you're determined, nothing stops you from downloading all versions of your GOG games straight into your Android phone. I'm not kidding. They won't work on Android, but then it's a matter of copying them straight to your PC.

This is in contrast to Steam, which will only let you install the games for the platform it's designed on. Steam on Linux will not let you download Windows games to use with Wine...you have to use Steam on Windows with Wine in order to get their games and run them with Wine....

Ah maybe it's true, but I would feel quite unsettled about the different editions possible incompatibilities.