I don't think you can increase speed of a motor to gain horsepower.
Yes you can, as long as you increase the voltage at the same time. Think of HP as a shorthand notation of "A certain amount of torque at a certain speed". If you increase either value, you increase HP. The trick is to increase one without a corresponding decrease in the other. That's what happens when you MECHANICALLY change speeds, i.e. a gearbox or belt and pulley system. Speed goes up but torque goes down, or vice versa, but HP remains the same (other than added losses in the process). But with a VFD, torque is a product of voltage AND frequency, you can think of it as a V/Hz ratio. If you increase or decrease them together at the same ratio, you can change HP without changing torque. Lower is of course easy, that's what we are essentially seeing with the traditional application of a VFD. At 50% speed we are at 50% voltage, torque remains constant because the V/Hz ratio is unchanged, therefore HP is 50%. But at 100% speed and 100% voltage, you are maxed out. Since you can't make the VFD put out more than 100% voltage, any further increase in frequency begins a DECREASE in torque; we call this "operating in Constant HP mode" because with no additional voltage, you can no longer maintain the V/Hz ratio and torque begins to drop off.
UNLESS... you can continue to increase the voltage as you keep increasing speed! That's where the trick employed by the OP's supplier comes in. It's a very common issue in high speed applications to wire a dual voltage motor at the LOWER voltage, but apply the higher voltage to the VFD. Then you program the VFD to supply 100% of the motor NAMEPLATE voltage (lower) at full speed (60Hz in our case). When you get to 60Hz and start to go beyond, the VFD has plenty of additional voltage to keep the V/Hz ratio constant as you go. So if you get to 2X speed, you still have 2X relative voltage (relative to the nameplate lower voltage as the motor is connected for). When you do this, the motor is now producing 2X the rated HP! Current is still the same as it was if it had been wired for the higher voltage, so the motor is not overloaded, although there are concerns:
1) Cooling fans in the motor may become less efficient at higher speed, it depends on the fan design.
2) The bearings must be rated for the higher speed.
3) Motor rotor balancing becomes much more critical.
4) LOAD considerations become very different (as you mention below).
But if you pay heed to these issues, it really is something that we do all the time, it's no big deal.
If you increase the speed of a driven load it will require more power to drive the load as more work will be done.
ABSOLUTELY in centrifugal loads like pumps and fans, in fact that's the big fallacy: it's rarely economical to do this trick on centrifugal loads because the speed increases the flow, and power REQUIREMENTS of the load increase at the CUBE of the flow increase. So for example if you increase the speed of a pump by 50% (150% of design), and you did the above voltage trick with a VFD so you have 150% of the HP now, your pump LOAD requirement on the motor will increase by 1.5 x 1.5 x 1.5 or 337.5%! So if you had a 10HP motor, increasing the speed by 50% makes it a 15HP motor, but now you need 34HP to make the pump work and you overloaded that motor. So if you want to do this for some reason, you have to START with a motor that is over 3X what you need so that you get the higher flow out of it when over speeded.
If this motor is a 4 pole motor the syncronus speed at 60 Hz is 1800 RPM. @ 100Hz it would be 3000 RPM. If anything the pump was probably designed to operate around 3000 RPM and motor was sized for load at that speed.
One would absolutely HOPE that the pump supplier who did this was aware of this exact issue, as I pointed out above. Unfortunately from my experience, many are not. I have repeatedly come across pump salesmen who don't totally understand this issue and ask me to "just turn up the VFD speed, that'll take care of it". Hopefully that is not the case in the OP's situation. But certainly it's worth investigating.
Again the motor experts can clue us in hopefully. I know higher frequency means less iron core is needed but not sure if less copper is needed. If so then you probably do increase horsepower capability of same motor by running at higher voltage and frequency. This is a reason for using high frequency on aircraft - to reduce weight.
Yes, that MIGHT be the reason they are doing this, if I recall from the OP (it's a long way back now), this was a submersible pump(?). Deep well submersibles have to deal with the issue of the weight of really long cables going down to the motor, so it might make sense to oversize the motor in order to get what you need with as little weight as possible, both for the motor and for the cable.
Again, I would hope that this is what this entire issue was all about, and that the pump OEM was/is aware of all of the ramifications etc. Some of the numbers posted in the original message actually don't work out correctly, but we don't know for sure if the OP was reading everything correctly either. Once you go "off nameplate" on something like this, it's hard to know for sure if something is an interpretation, typo, assumption or just plain incorrect.