That's a good question actually.
The data for your online game and stats etc is stored only on your console I'd assume. What I noticed a long long time ago was if you go to your save data on the Wii, games that have online play generally have a larger save which could be the extra data being stored. You also aren't supposed to be able to copy these saves since they have data that is supposed to be unique for one player, but by turning on the 'No Copy Protection' flag in priloader you're able to copy them to and SD card to another Wii.
Which leads to: What would happen if you had the same exact save on 2 Wii consoles, and tried to use them both?
Well... I don't know for a fact

All i can assume is that you'll play the first save as much as you like with no problem. When you move onto the second save, your leader board scores would be updated with the lower less progressed scores. You wouldn't have the unlockables since you haven't got them on that save. If you went back to the first save again and were still further progressed then the leaderboard would revert back to the higher score hahaha
Using them at the same time... Maybe a ban? There's definitely something not right when they get the exact same information coming in at the same time, but can't be sure on that one
As for leader boards, there will be some kind of database that stores the scores of everyone worldwide, so they're not part of your save game. For example, many games ask if you'd like to upload your score/ time to a leader board, it'll just be sending the data to a database and placing you in chronological order depending on how well you did.