On the other hand batteries energy capacity is commonly rated in Watt hour.

Watt*day is a perfectly cromulent energy statement although a bit misleading.

Watt*day is weird, but at least unit of energy. Watts a day is not.

Really common mistake in general to use kWh as a kW. Watthour is unit of energy. As watt is energy by time period. So you get back to units of energy.

Technically watt/day could be change in power consumption.