Some PC games, when you install them, afterwards, thay ask you to input a CD key, before you can start the game up, this is not the case with D2, you install the game and launch it, it doesnt ask for a key.

Maybe you should contact Larian Support :

support@larian.com

Outline your problem and ask them if there is any version of the game that requires a CD key.

Hope they can help you progress this further.