Medieval man definitely had the same genome, but not the same body. Factors such as availability of good food influence your growth. Medieval people were, compared to us, rather small, which can be seen from both their skeletons and various objects of daily use (height of doors, armors).

This indicates insufficient nutrition for most of the population. Famines were common approx. 5-7 years, so if you were one of the small folk, you experienced two to three of them as a juvenile.

And bad nutrition has some effect on brain, too, after all it is part of the body. I do not want to claim that medieval people would be unable to learn calculus, but those with worst food security might struggle.

But isn't that also environmental? Diet of the surrogate mother and child growing up over the years?

Oh, yes, absolutely. Subsistence farming in mild climate isn't a great and healthy lifestyle for an ape which evolved as a hunter-gatherer on the equator.

I also expect that a large part of the average intelligence that has changed would be due to improved nutrition.

It recently (last few generations) went down and then up again because we thought it'd be a great idea to give the whole world lead poisoning.

Just wait until you see the effects of >400ppm atmospheric CO2 will do to the distribution of arable land, and thus the effects on those species which depend on agriculture to survive.

Yeah, nutrition is listed as one of the causes for the flynn effect: https://en.wikipedia.org/wiki/Flynn_effect

The flynn effect has slowed down in the western world. This highly suggests not genetic changes but genetic potential being fulfilled thanks to environmental improvements.

That was only an example to show that environmental influences start very early. Your environment continues to influence you until you die, even if that influence in your final years is of a supportive nature. Many elderly people can live quite well in their homes they lived in since decades but once they come to a nursing home, they often become disoriented.

And yes, medieval people were as intelligent as modern people. Many of the smartest minds of humanity haven't lived today but hundreds or thousands of years ago. I'd put all behaviorally modern humans are into the same category.