LEDs are light-emitting diodes, and a diode is an electronic component through which electricity can only go in one direction. In electrics, the anode is the side that electricity comes from, the cathode is the side it goes to.¹
So, with two anodes and one cathode, the side of the LED that lights up is the one that you connect to the + side of the power supply. If you connect both anodes to the + (and the cathode to the −), I would think both colours of the LED will light up. With a switch in the cathode wire, you can turn both on or off at the same time, while with switches in the anode wires, you can turn them on and off independently.
OTOH, in an LED with two cathodes and one anode, you need one wire from the + to the anode and two wires from the cathodes to the −. The net effect will be totally the same, though: you can make one or both sides light up, depending on where you put the switch(es).
¹ Though remember here that electrons, which are what electricity actually is in a metal wire, go the other way — that is, they flow from the −-side of the power supply to the +-side.