Can the iCloud share app-specific data across devices?
The direct answer to your question is It depends.

Certain applications like iWork can store data in the cloud which is then synchronized across devices. However there is currently no interface to allow you to choice which document is the primary, as the data is synchronize on a last updated cycle.

If the game support using shared document via iCloud, you would be able to synchronize game data, however, by default it will not.