Hi Mudell….To answer your question…No…they will last the same amount of time. But I think there maybe something you are misunderstanding.
Without getting distracted with all sorts of clever comments about minor variations in battery physics…etc, and by just focusing on basic electrical theory, maybe the following will help.
Power = Voltage x Current ( P= E x I)
Your Battery 1 and 2 specifications fulfill the formula. Consequently they supply the same power to the load, ie 10Kw.
The current draw from a load consuming the same amount of power to operate it, is dependent on the nominal voltage to drive that load. You could have two separate water pumps rated at 500W. If Pump 1 had a nominal input voltage of 450V (same as Battery 1) then it would draw 1.11A only (P=E x I or if translated, I = P / E). The pump would run for 20hrs off your 450V - 22.22A/Hr Battery.
Pump 2 on the other hand might have a nominal voltage 48V (Same as Battery 2) and considering it is also rated at 500W, it will draw 10.42A. Again same basic formula (P=E x I or if translated, I = P / E). The pump would run for 20hrs off your 48V - 208.33A/Hr Battery.
The term 10KwH is the most inappropriate term used to rate the capacity of a battery. It stores 10Kw of power, and if you have an appliance that consumes 500W, then that battery will last 20hrs. That is the basic thing you need to understand. Nominal voltage of the battery and appliance must match, and that then dictates the current draw.