Well, the biggest issue here are traditions.
In general, when we talk about the battery capacity, we should internally think about the amount of energy which is stored inside of the battery, since electrical devices consume energy to work.
Amount of energy is measured in joules. As unit of power, watt is one joule per second. Therefore, if we multiply watt value by a unit used to measure time, we get energy. So one watt*hour is 3.6 kJ.
On the other hand, we have the ampere*hour measurement unit, and for portable devices, its derivative milliampere*hour.
As you noted in your question, we need to multiply the ampere*hour value by battery voltage to get the watt*hour value, and we've already established that the watt*hour is a measurement unit for amount of energy.
Therefore, to properly compare run-times using different battery types, you need to use the energy values, that is to say the watt*hour values. Since there are different "standard" laptop battery voltages, you need to compare them using energy that they store, which is what your software seems to be doing.
This still leaves us with a bit of confusion about famous "mAh" numbers we often see on batteries.
The idea of its use is as follows: My device uses the battery's nominal voltage, and it needs current of x mA to function. Therefore, if I power it with a yy mAh battery, my device will work for yy/x hours.
However, for this approximation to work, you need to fulfill the assumptions. Pretty much all phones today use a single cell lithium based battery, with nominal voltage of around 3.8 V. Therefore, it's useful to use mAh to compare such batteries. Cars often use batteries with nominal voltage of 12 V, and Ah is used to compare them. However, when you're trying to compare two dissimilar battery types, you can't use the Ah values, since the battery voltages are different.
To finally answer your question: If you had a way to lower voltage of your laptop battery to the voltage your phone needs to work, and you could do that without wasting any energy, then the laptop battery would last longer, since it has ~3 times more energy than your phone battery.
In real life, we could achieve efficiencies in the 80% to 90% range, perhaps even a bit more.
If you were, however, to just turn the extra voltage of laptop battery into heat, then the phone battery would last longer.