Trynn, Champion of Freedom

"Partner with [name]" represents two abilities. The first is a triggered ability: "When this permanent enters the battlefield, target player may search their library for a card named [name], reveal it, put it into their hand, then shuffle their library."

The triggered ability of the "partner with" keyword still triggers in a Commander game. If your other commander has somehow ended up in your library, you can find it. You can also target another player who might have that card in their library.

A creature with a "partner with" ability can't partner with any creature other than its designated partner. Losing a partner ability during the game doesn't cause either to cease to be your commander.