Although there is no explicit rule for it, treating decks with different assigned commanders as different makes the most sense.
The choice of commander is certainly a property of a commander deck because its legality depends on it, but it is a matter of individual tournament rules whether that matters as far as switching commanders between games is concerned.
Ultimately, it is most important to adhere to the core philosophy of the game mode:
Commander is designed to promote social games of magic.
It is played in a variety of ways, depending on player preference, but a common vision ties together the global community to help them enjoy a different kind of magic. That vision is predicated on a social contract: a gentleman's agreement which goes beyond these rules to includes a degree of interactivity between players. Players should aim to interact both during the game and before it begins, discussing with other players what they expect/want from the game.
House rules or "fair play" exceptions are always encouraged if they result in more fun for the local community.
So whenever there is uncertainty about a certain rules question specific to the Commander mode, talk to the people in your play group and reach an agreement.
That being said, there are hints on how to resolve the question with rules. A Commander-legal deck is required to assign a certain card as its Commander, and if any rules are not covered by the Commander section, the regular rules apply:
903.1. In the Commander variant, each deck is led by a legendary creature designated as that deck’s commander. The Commander variant was created and popularized by fans; an independent rules committee maintains additional resources at MTGCommander.net. The Commander variant uses all the normal rules for a Magic game, with the following additions.
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.
A card that is defined as the commander of a deck is different from another card with the same name that isn't a commander. Therefore, two decks containing otherwise identical cards but different commanders are overall different decks.
For single-game matches, this doesn't matter. You simply declare another card as the new commander between games and play accordingly.
For "Best of N" matches, or during tournament play, modifications of the deck are not allowed except as outlined in the Sideboard rules or if a card like Burning Wish allows you to. In those situations you could not swap out the commander between games.
Another thing to consider are the remarks from the maintainers of the Commander mode, section "House Rules":
D. League Play
A League is a regular group of players who maintain the same decks across multiple play sessions. Within a given league, Commander are allocated first-come, first-serve and are preserved between meetings/games. No player may have, in his or her deck, the Commander of any other player in the game; it should be replaced with some other card before the game begins.
The spirit of this rule seems to be that decks, including Commanders, remain unchanged.