Though it doesn't relate to purchase and registration of the game, another complication you should be aware of in regards to playing on multiple platforms is that some configuration settings (your HUD layout, hotbars, and gearsets, along with a few other settings) are stored locally on your client machine. So if you change one of those on one platform, then next time log in from the other platform, those settings won't automatically carry over.
I think you can manually copy some setup files across between platforms to transfer those settings, but I've only actually heard of people doing that when they played on two different PCs (typically a desktop and laptop). Mac isn't as widespread, but I'd assume it would use the same configuration files, in which case copying them across would work between PC and Mac the same way. (Leaving just those who alternate between consoles and computers left out and needing to re-do all such settings from scratch, since consoles do store things differently.)
