Power and torque both have an effect on the performance of a car.
It is possible to increase the power without increasing the torque, just by desiging the engine to rev higher.
Power = torque*revs
When engines are designed, their output characteristics are mapped to suit the intended application.
This might mean that a car is economical, or that it has very high total output power, or that it produces a lot of torque at a certain engine speed.
One way to look at the way power is produced is to plot the power and engine revs on a chart and observe the curve.
This can also be done for torque and revs too.
When you then look at both these charts you can get some idea of the overall performance.
However, the final characteristic depends also upon the engine gearing and transmission losses too, and you can modify the gearing to suit the engine better.
Finally, to answer the OP, if you were to take a vehicle and increase the power by 100%, you would run into problems, the structure of the vehicle is not likely to be able to take this without some modifications,
The clutch would need to be uprated, the gearbox would probably need to be heavier duty, the differential would need to be heavier duty, the engine mounts might not be strong enough to withstand the torque reaction from the engine, and tyres and brakes would probably not be good enough.
This would all mean a heavier vehicle.
If you intended to increase power and see what the effect was, it would also depend upon how exactly you increase the power.
If the engine produces more power by revving higher, then what would happen is that it would not accelarate any faster, but it would accelarate for far longer.The reality is that modifying an engine to rev higher often means that lower rev power is sacrificed to some degree and so accelaration can actually be slower.
If the engine produces more power by increasing the torque, then it will accelarate faster, up to the original rev ceiling.
Increasing the torque may well not increase the vehicle speed by much at all, because the speed is determined by the engine revs through the transmission chain, the engine would still top out at the same revs, however with more torque available it would be possible to change the gearing ratios and this would allow a higher speed.