A UPS is not an energy storage device, it is a sort of conversion device. A UPS takes energy from its storage bank (the battery) and converts it into a different form, ideally, a UPS can keep doing this indefinitely long provided its reservoir is full. So, there is no point in defining a UPS in terms of Ampere hour or any storage quantity. But a UPS is limited by the RATE at which it can supply energy and energy per unit of time is basically power, so it makes sense to specify a UPS in terms of a unit of power.
UPS is rated in VAs (or in Watts). The output VA or watts means how much power that electronic box is able to provide. If you put a load that is higher than this (for example putting a 750W load on a 650VA UPS), the UPS fuse will blow out.
Whereas a battery is a storage device (charge storing). So, it makes perfect sense to describe a battery in terms of how much charge it can store. The amount of time the battery will last depends upon the rate at which charge is extracted from it. A battery holding 10 C of charge will last for 10 sec if it supplies at the rate of 1C / sec whereas it will last for 5 sec if supplies 2C/sec and so on. 1Ah = 3600 C
In short:
UPS is rated for capacity for converting dc voltage to ac(V) with power rating(A), like a number of lights, fans, etc. The battery is for storage capacity, of power(A) for time(H), and how long power last.