I think the idea is that watt-hours (Wh) serves the same purpose and doesn't miss information about capacity if voltage is omitted. Technically though the units in watt hours should cancel out to be Joules which may be the best unit.
Are amp hours in some way better due to the varying voltage across the charge profile of a draining battery perhaps?
I'd argue that the undecomposed watt-hour is in fact the superior unit for batteries.
Because what I want to know about a battery is exactly how many hours I can use it at a given wattage. The voltage is determined by application, and the only amp-related stat I'm interested in is maximum current, which is more useful to me in watts anyway.
So a 50 Wh battery which I'm going to draw at 10 watts? five hours of use time. 20 watts? 2.5, and I can do this in my head all day. Joules are less convenient because going from seconds to hours is too annoying to do as mental maths.
It's mostly because the voltage is assumed to be constant across the discharge profile (even if it isn't), and Ah is taken to the point that you can safely discharge the battery to. So you compare two batteries with the same voltage on their Ah, and systems in terms of Wh (and then the voltages are irrelevant).
One problem here is that what looks like a battery on the outside is often a system under the hood, so the output voltage is not necessarily the voltage at which the battery operates due to for instance DC-DC convertors and other electronics embedded in the battery housing.
It's only useful if you know the voltage of the battery, so that you can calculate watt-hours, or if you at least know that two batteries you're comparing are the same voltage so you're comparing like to like. It's not useful if you're comparing batteries of different voltages unless you're adjusting for voltage in which case you want to use Wh anyway, which is why using Wh here is more useful.