Blue, Loyal Raptor

Once the game begins, your two commanders are tracked separately. If you cast one, you won't have to pay an additional {2} the first time you cast the other. A player loses the game after having been dealt 21 damage from one of them, not from both of them combined. Command Beacon's effect puts one into your hand from the command zone, not both.

If Blue, Loyal Raptor enters the battlefield with one or more kinds of counters on it at the same time as other Dinosaurs you control, those other Dinosaurs won't get additional counters from Blue, Loyal Raptor's last ability.

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, whether or not they have that card in their library.

"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 second ability represented by the "partner with [name]" keyword modifies the rules for deck construction in the Commander variant and has no function outside of that variant. If a legendary creature card with "partner with [name]" is designated as your commander, the named legendary creature card can also be designated as your commander.

Note that the target player searches their library (which may be affected by effects such as that of Stranglehold) and that the card they find is revealed, even though these words aren't included in the ability's reminder text.