Electrical Motors and Frequency
I like your thinking on this. You see something about it that bugs you so you ask a question. Those little things that bug you because they don't fit, and the curiosity to pursue them are very positive traits in engineering and science.

But I think you are mixing up a few different motor types. Induction motors are the ones that have slip. If you run an induction motor at a fixed input frequency with no load it will spin up until it ALMOST matches the electrical frequency. There will only be a tiny bit of slip. When you add a mechanical load, three things happen at the same time to balance out the physics of it:


the motor slows down (increasing slip)
the current going into the motor increases (increases input power)
the output torque of the motor increases (increased output power)


At some point the motor will find a new equilibrium with the changed load (assuming the load does not overwhelm the motor or something like that).

Changing the frequency is similar. In your scenario, you have a motor in steady state, and you increase the frequency by a small amount. Instantaneouslythe slip, torque and current will increase, balancing out all the terms for conservation of energy. Of course motors are less than 100 percent efficient, so the output power will always be less than the input power. Also, current * voltage is power, but you also need to consider power factor (search power factor if interested...). We are glossing over a lot of details.

But there is a bit of a catch here. For an induction motor, if you change the frequency more than a little bit, you really want to change the voltage in unison. For example, if the motor is happy running at 240V and 60Hz, it will also be happy at 120V and 30Hz or 60V and 15 Hz. As long as the Volts over Frequency (V/F) remains at that ratio of 4, the motor will be happy and will be able to supply its rated output torque.

There are a lot of exceptions and qualifications to what I said here. For the most part it doesn't apply readily to single phase induction motors. And anyway this is the sketch version. If you need to actually get into motor drives there are a lot more details to learn about.

Some cars do use three-phase induction motors. Their motor controllers are very sophisticated and highly optimized for power efficiency.

But other cars may use BLDC motors or permanent magnet synchronous motors. Everything is pretty much the same except that slip must be maintained at zero. The motor controller must maintain the rotating electrical field produced by the stator in phase with the rotor.  But when the load increases, the motor drive control will either reduce the frequency and voltage to allow the motor to slow down or it will increase the voltage a bit to compensate for the increased load and maintain constant speed (if that is what it is trying to do). If the controller is maintaining constant speed, then when the load increases, the current will also increase to compensate. The voltage may increase a tiny bit but not as much as the current. And if the controller wants to speed up the motor it will ramp up the voltage and frequency.