I can think of a few possibilities here:
A taller pot will hold more moist surface near to the surface of the liquid, and may slow down the evaporation rate when you’re simmering. For materials that conduct heat well, if the pot above the liquid is significantly hot it might be acting as a heat sink, transferring some of the heat to the air and not just the food.
A wider pot will concentrate the heat differently. This can be especially significant with electric stoves (which does not have heat lick up the sides like with gas heat). This results in the sides staying cool, especially for cast iron or other heavy pans, which can change how something cooks.
Different materials will heat up differently. Aluminum is a pretty good conductor so should transfer some heat up the sides, such as a tri-ply pot (and unlike a stainless steel pot with a copper or aluminum disk on the bottom). But you also said that the aluminum was dark and so it will both absorb and radiate heat easier than a light, shiny surface.
When changing pots for things that you have perfected, I would suggest looking at the signs of how something cooks— is it bubbling the same? Is it warm along the sides of the pot? (Don’t check this with your hand for small pots on gas stoves)
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Ordinarily, I’d recommend adding a lid to try to balance things out, but you require evaporation, so that won’t work. You’re likely going to need to experiment with adjusting the heat to see if you can come up with the same result… and it’s possible that you can get close (and maybe even better), but it will never be exactly the same.
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And now for a thermodynamics lesson:
Assuming that your stove’s burners put out constant heat, equilibrium is when the system is shedding an equal amount of heat. Some of this might be heat that is never absorbed by the pot or radiates from the pot into the room and may not all go into state changes (evaporation, creation of new chemical compounds, etc)
If you want to do a real test, you’re going to need a long thermometer, and check to see not only what the temperature is at the middle of the pot but at the bottom and near the surface as there will be a temperature gradient.