Why are batteries measured in ampere-hours but electricity usage measured in kilowatt-hours?
\$kW \cdot h\$ are a measure of energy, for which grid customers are billed and usually shows up on your invoice in easily understood numbers (0-1000, not 0-1 or very large numbers; ranges which, unfortunately, confuse many people).

\$A \cdot h\$ are a measure of electrical charge. A battery (or capacitor) can store more or less a certain amount of charge regardless of its operating conditions, whereas its output energy can change. If the voltage curve for a battery in certain operating conditions are known (circuit, temperature, lifetime), then its output energy is also known, but not otherwise, though you can come up with some pretty good estimates.

To convert from \$A \cdot h\$ to \$kW \cdot h\$ for a constant voltage source, multiply by that voltage; for a changing voltage and/or current source, integrate over time: $$ \frac{1 kW\cdot h}{1000 W\cdot h}\int_{t_1}^{t_2} \! I(t)E(t)dt ~;~~E~[V],~I~[A],~{t_{1,2}}~[h]$$