Why battery specs are given in contextless mA·h instead of meaningful W·h?
If you assume that a battery or power supply is a perfect voltage source, than the two are equivalent. Multiply capacity in milliamp-hours by the battery voltage to get capacity in milliwatt-hours, or multiply current by input voltage to get power in watts.

For many devices, the current (mA) will be the same regardless of input voltage. This is true of anything with a linear voltage regulator; current remains constant, but as input voltage goes up, power (W) increases because it is the product of current and voltage. The linear regulator simply converts excess voltage into heat.

For batteries I don't have a good excuse. We could also give capacity in watt-seconds or joules (same thing, a unit of energy), but that would introduce other problems. I suppose a battery is a complex device, and the available energy depends on the load (high current? low current?), temperature, and many other factors. In the absence of a simple and accurate model that covers all cases, the convention has favored units that make common calculations convenient.