If a 14 ga wire is good for 15 amps @ 120 volts it will only be good for 15 amps at any voltage. voltage rating on wire is a measure of the insulation not the wire. it is how much voltage the insulation will withstand before it breaks down and shorts out.
Current is the amount of electron flow a wire can stand before it breaks down due to heat.
You can do a lot of things with wire but to be safe you should always stay below the current rating of the wire no matter what the voltage is.
Your reasoning about 240 having 2 legs is flawed some what since you are not taking into account that for a fixed load the current will be somewhat divided between the two legs but not at a 50% rate. i can give you more information on how this works but I can say that if a wire is rated for 240 it will not carry twice as many amps as the same ga wire that is rated for 120volts.
I was with you jack and then you went into the ditch. The current in the two-pole circuit breaker leaves one pole of the breaker and enters the other pole to complete the circuit. There is no 50-50 split or any other such voodoo.
Example: 120 volt @ 15 amps, single phase, # 14-ga wire, one pole breaker.
120 x 15 = 1,800 watts. Maximum power attainable.
With a 4% voltage loss # 14-ga wire at 25-foot max length.
120 x .94 x 15 = 1,692 watts
# 10-ga wire will have the same loss at 50 feet.
Same problem; 240 volts @ 15 amps, single phase, two-pole breaker.
240 x 15 = 3,600 watts. Maximum power attainable.
With a 4% voltage loss # 14-ga wire at 50-foot max length.
240 x .94 x 15 = 3384 watts
# 10-ga wire will have the same loss at 75 feet.
You can see that doubling the voltage will double the power for the same wire gauge and increase the circuit distance.
There are other NEC tables taking other conditions into account such as insulation temperature range and operating temperatures.