Why is electricity then measured in kW/h?

In my understanding, power (instantaneous) is measured in W, but consumption needs to be integrated over time, thus the per hour part.

It's not kW/h, it's kWh https://en.wikipedia.org/wiki/Kilowatt_hour

Which is kW * h, i.e. kilowatt times hours, like man-hours are number of men multiplied by time.

kW/h would be a unit of the derivative of power, measuring the rate at which power consumption changes.