As far as I can see, you can't buy an avatar in the mobile game but you can in the PC version which screams out to me that if it's bought on the PC version, it will only work on there.
I think some of you would be surprised at how many are using mobile, go look at the amount of players in the coin rooms, add to that those playing other parts of the game eg CC comps and there are a significant amount of us. This is a new and lucrative revenue stream for WGT and I have no doubt that in future, anything that makes them money, if it can be ported to mobile, it will be.