Why is electricity consumption billed in watts and not amperes?

They don’t bill in watts (power) either. They bill in watt-hours, that is, energy consumed. (kilowatt-hours typically.)

Let’s break this down a bit. Current alone doesn’t tell you power. You also need to know the voltage, as power is voltage * current. Then, you tally power over time to figure energy.

Now you could estimate power from current if you make an assumption about the voltage, and in the early days of electricity they did exactly that using ‘coulomb counters’, that is, they measured and tallied the current delivered over time and billed based on the tally.

This current-only method proved to be inaccurate because of line fluctuations, so for this and other reasons (notably, the adoption of AC power) they developed the motor-type meter, and later, the more familiar spinning-disc induction-type watthour meter. These meters also take voltage into account by design.

More about meter development here: https://www.smart-energy.com/features-analysis/the-history-of-the-electricity-meter/

And because it matters sometimes, a discussion of real vs apparent power and how utilities deal with it: https://www.electronicdesign.com/power-management/article/21806945/how-does-power-factor-correction-impact-your-utility-bill

Why would companies bill for wattage instead of amperes?

Because amperes don't tell the full story about energy transfer from a source to a load. If you supplied a load that took 100 amperes at 1 volt, the power consumption (joules of energy per second) is 100 watts. If a different load took 100 amperes at 100 volts, the energy transfer per second is 10,000 watts.

If you only billed in amperes you bill both customers the same.

It's all about energy and power so, you calculate power delivered to the load and that accumulation of power with time (energy) is what you are billed on. To calculate power it is amperes x volts.

I think because they care about energy consumption, energy is what costs money and resources to generate, also you can have different voltages when using 2 and 3 phases which would make the charge measurement by itself useless

-- edit, and yes, they measure in watt*hours for obvious reasons