They would not track separately; the player will lose.
903.10a A player that’s been dealt 21 or more combat damage by the same commander over the course of the game loses the game. (This is a state-based action. See rule 704.)
903.3. Each deck has a legendary creature card designated as its commander. This designation is not a characteristic of the object represented by the card; rather, it is an attribute of the card itself. The card retains this designation even when it changes zones.
Basically, no matter who controls a commander, it is still the commander of the deck it started in; it is the same actual commander. While you can take control of a player's commander, there is no way to change it into your commander.
In regards to your question about 1 player controlling 2 separate commanders, this can easily happen with the Partner ability, that allows you to have 2 Commanders. In this case, damage from each commander is tracked separately; you need 21 damage from the same commander to lose the game.
702.123d Except for determining the color identity of your commander, the two commanders function independently. When casting a commander with partner, ignore how many times your other commander has been cast. When determining whether a player has been dealt 21 or more combat damage by the same commander, consider damage from each of your two commanders separately. See rule 903.11a.