I expect that the answer to the question in the title is "yes", but I'm not sure where the official rules for Commander deck construction are.
Is a chosen Commander a property of the deck, or is it possible for a single Commander deck to contain multiple possible Commanders and to choose before each game (as long as each meets the general requirements such as colour identity)?
It doesn't matter. Since you can change decks between games, you can change to a deck that's identical except for the choice of Commander.
Everyone talks about the Commander as if it's a property of the deck, including the Comprehensive Rules and the EDH site.
CR: In the Commander variant, each deck is led by a legendary creature designated as that deck’s commander.
EDH: Players choose a legendary creature as the "Commander" for their deck.
But that's not necessarily significant. It's simply the obvious and simple way to explain Commander. There's no explicit mention in the rules that the Commander is a property of the deck.
Does it matter?
In a casual setting, one may change decks between games as often as one desires, so you can obviously play two games with decks that vary only by your choice of Commander. The answer to your question is therefore irrelevant in a casual setting.
The answer is only potentially meaningful in a tournament setting, but I can't find anything suggestion that DCI hosts Commander tournaments. That means the the answer to your question is irrelevant as far as Wizard-sanctioned tournaments go.
That leaves non-sanctioned tournaments. Since they're non-sanctioned, there's no way to predict their deck registration policy. The answer is unanswerable for those in general.
The only tournament I was able to find is this league. But since they allow you to change deck between games, the answer to your question is irrelevant there too.
Given all of the above, my answer to your question is as follows:
If you enter a situation where you must reuse a deck for more than one game, consult with those with which you enter into this situation on this matter. Otherwise, it doesn't matter. Since you can change decks between games, you can change to a deck that's identical except for the choice of Commander.
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:
- Commander
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.
When you start the game, the card chosen as your commander must meet the color identity of the deck (well, vice versa, but the point is similar).
This is pretty much a given from wizards if you look at the Commander 2013 products. In those decks, there are 3 commander oversized cards for the commanders you could choose, with the other potential commanders staying in the deck.
Given all that, if you change your commander in your deck, it is a new deck, for all intents and purposes. Now, in a tournament setting or a Best-of-X setting, is that kind of deck change allowed? I dunno, ask the host.
In most commander sets released the last couple of years there were multiple legendary creatures in the same colour identity as the commander. those creatures could be used as an replacement for the initial commander if you didn't like the initial commander.
But to use an example of my own aurelia commander deck. It is build around the premise that aurelia is the commander and that I have angels as theme in the deck. But i could use my Iroas in the deck as a commander because it has the same colour identity.
As example of the commander decks released by wizards. In commander 2013, each deck had three oversized commander cards which are creatures you can use as a commander.
There isn't a universal rule about choosing to change your commander in between games, that I can identify. As long as the card you designate as your commander at the start of the game, per rule 903.6, meets the requirements of matching the color identity of your Commander Deck, it is a valid selection.
903.6. At the start of the game, each player puts his or her commander from his or her deck face up into the command zone. Then each player shuffles the remaining 99 cards of his or her deck so that the cards are in a random order. Those cards become the player’s library.
Tournaments out in the world could place a restriction upon your deck that the Commander always be the same card. They could also permit changing your commander in between games.
Presumably, any legendary creature you put in your Commander Deck that has the same color identity as your designated Commander would be a legal choice to designate as the commander. The next question becomes, why would you want to?
The rules governing commanders are as follows:
903.1. In the Commander variant, each deck is led by a legendary creature designated as that deck’s commander.
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.
903.6. At the start of the game, each player puts his or her commander from his or her deck face up into the command zone. Then each player shuffles the remaining 99 cards of his or her deck so that the cards are in a random order. Those cards become the player’s library.
As you can see there isn't anything about a deck only having a single commander. Also as has been mentioned the Commander decks put out by Wizards usually have 3 options to use for the commander.
Tournaments or leagues can changes these rules, but as Commander is not a format that can be sanctioned the MTR doesn't apply. Thus they are allowed to create their own rules for changing commanders between games.
