You can choose two commanders with partner that are the same color or colors. In Commander Draft, you can even choose two of the same commander with partner if you drafted them. If you do this, make sure you keep the number of times you’ve cast each from the command zone clear for “commander tax” purposes.
To have two commanders, both must have the partner ability as the game begins. Losing the ability during the game doesn’t cause either to cease to be your commander.
If your Commander deck has two commanders, you can only include cards whose own color identities are also found in your commanders’ combined color identities. If Falthis and Kediss are your commanders, your deck may contain cards with black and/or red in their color identity, but not cards with green, white, or blue.
If something refers to your commander while you have two commanders, it refers to one of them of your choice. If you are instructed to perform an action on your commander (e.g. put it from the command zone into your hand due to Command Beacon), you choose one of your commanders at the time the effect happens.