Don't know if the title makes sense so I'll explain what I'm doing. I'm trying to wire two adjacent fridges that each require a dedicated 15A circuit. I'm just wondering instead of running two wires if I can run a single 14/3 wire with the black wire powering one fridge, the red wire powering the other but both using white wire as neutral. My instinct says it's not up to code but I thought if ask. Thanks!
Pay very close attention to that requirement "that you must use a two-pole or handle-tied breaker".
More specifically, this means that these are NOT two breakers on the same phase; this is a split-phase double breaker with the two lines on the opposite phases. You have 120V from each hot (black and red) to the neutral, and 240V across the two hots (which you aren't using in this application).
The shared neutral is allowed because the two phases are flowing in opposite directions, so to speak. As one is flowing one direction, the other is flowing the opposite, so you get some cancellation effect.
The common will be carrying the current for both fridges, and, therefore, will not provide the same current carrying capability as that expected for the gauge of the wire. This is done in ceiling fans which also have lights on separate wall controls, but they don;t draw as much current as fridges. Make sure you use a gauge of wire that can carry the total current of the two fridges on the common; probably 10 gauge in your case. All that said, if the fridges are on opposite sides of a 220 box, you might get away with it, as explained above.