Last time I checked, 2 times 10 was 20. I don't know how 17.3 come into this discussion. So no, the math is not correct.
If the dryer is rated for 2300 watts, then the current at 230 volts would be 2300/230, or 10 amps. So that is what I think the rating of 10 amps is based upon. But this would be a resistive load, so the power it will deliver at a lower voltage will not be 2300 watts. At 208 volts, the current would be (10) times (208/230), or about 9 amps. So #12 wires should be fine. Just to complete the story, at 208 volts the power delivered will be (2300 watts) times (208/230)-squared, or about 1880 watts. Are you sure the owner will be happy with that? Perhaps a different model of hand dryer might be a better choice?