Dominion is balanced largely through the fact that it's symmetrical. No matter how good or bad the kingdom cards, both players are interacting with them in the same way, so, barring things like writing cards that give additional advantage to the first player, it's hard to unbalance it.
If you played "a game like Dominion, except each player had their own independent set of Kingdom cards," it would be big-time not balanced. Like, kingdom sets aren't balanced against each other.
Or perhaps you meant that it's balanced like "a given card tends to be about as good as a given other card." But that's just... not very true. In any Kingdom, there are typically a few must-buys and a few must-not-buys, and there are cards that are almost never good buys and cards that are almost always good buys.
Source: played a few thousand games of Dominion on isotropic back when that was a thing.
Disclaimer: Haven't played Dominion since Hinterlands, so this may not still be accurate.
If you played "a game like Dominion, except each player had their own independent set of Kingdom cards," it would be big-time not balanced. Like, kingdom sets aren't balanced against each other.
Or perhaps you meant that it's balanced like "a given card tends to be about as good as a given other card." But that's just... not very true. In any Kingdom, there are typically a few must-buys and a few must-not-buys, and there are cards that are almost never good buys and cards that are almost always good buys.
Source: played a few thousand games of Dominion on isotropic back when that was a thing.
Disclaimer: Haven't played Dominion since Hinterlands, so this may not still be accurate.