A single phase air conditioner takes 8 amps, while the same with 3 phase takes only 3 amps, the billing is finally for the amps consumed right, so why not only 3 phase.

The billing is usually in watts. A three phase that uses only 3 amps isn't necessarily more efficient, than a single phase that uses 8 amps.

For three phase you can calculate by the following:

.208 kV * 3 A * √3 = 1.08 kVA

If you have to account for PF correction, then that will affect your wattage say up to 15% difference.

For single phase you will find it looks something like this:

8A * .120kV = 0.96 kVA.

It seems to me that you will be billed for less wattage with single phase than three phase, on top of that in some cases you have to pay additional fees for a three phase hook up, and sometimes in residential situations, they charge extra per KWH for three phase power.

The difference lies in how three and single phase work. With three phase you're getting power from three different lines, propelling the motor every third turn. This usually makes three phase more efficient, especially when you need powerful motors. Single phase either uses brush-less (which consumes slightly more) or is only able to propel it every half, or single turn. This means to get the same power it has to push harder more often.

The harder a motor needs to be pushed, the more amps it consumes... a 120v single phase needs to draw more amps to achieve the same amount of torque a 208v three phase motor requires.