Looking for a story about a post-apocalyptic earth where an AI learns about humanity

Question

Eventually grows a person in a lab and transfers his consciousness, and so does a/the other AI present on the Earth.

Answer 1

"For a breath I tarry" by Roger Zelazny, 1966 (ISFDB). Published among others in "The last defenders of Camelot"

For ten thousand years Frost sat at the North Pole of the Earth, aware of every snowflake that fell. He monitored and directed the activities of thousands of reconstruction and maintenance machines. He knew half the Earth, as gear knows gear, as electricity knows its conductor, as a vacuum knows its limits. At the South Pole, the Beta-Machine did the same for the southern hemisphere.

The story features two AIs, Frost and the Beta machine, in control of infrastructure and maintenance on the northern and southern hemisphere. There are two more AIs, Solcom and Divcom, tasked with overseeing the whole process. Humanity has been wiped out, but Solcom and Divcom blindly follow their orders, both asserting that the other does not have the proper authorization and both destroying the others works whenever they can.
Frost is a loyal agent of Solcom.

Divcom sends another AI, Mordel, that taunts Frost with the fact that he does not really understand humans and the intentions behind their orders.

“Regard this piece of ice, mighty Frost. You can tell me its composition, dimensions, weight, temperature. A Man could not look at it and do that. A Man could make tools which would tell Him these things, but He still would not know measurement as you know it. What He would know of it, though, is a thing that you cannot know.”

“What is that?”

“That it is cold,” said Mordel and tossed it away.

...

“I told you that Man possessed a basically incomprehensible nature. His perceptions were organic; yours are not. As a result of His perceptions He had feelings and emotions. These often gave rise to other feelings and emotions, which in turn caused others, until the state of His awareness was far removed from the objects which originally stimulated it. These paths of awareness cannot be known by that which is not-Man. Man did not feel inches or meters, pounds or gallons. He felt hear, He felt cold; He felt heaviness and lightness. He knew hatred and love, pride and despair. You cannot measure these things. You cannot know them. You can only know the things that He did not need to know: dimensions, weights, temperatures, gravities. There is no formula for a feeling. There is no conversion factor for an emotion.”

“There must be,” said Frost. “If a thing exists, it is knowable.”

“You are speaking again of measurement. I am talking about a quality of experience. A machine is a Man turned inside-out, because it can describe all the details of a process, which a Man cannot, but it cannot experience that process itself as a Man can.”

At the end of the story (spoiler) ...

... Frost recreates a human, transfers his consciousness, and finds out that he now posseses emotions like fear. He learns that his commands will be obeyed by other machines. He sends a note to the beta machine, inviting it/her to join him as a human female, to start the human race anew.

Other plot details I remember: There is a rampant digger machine running over the earth, the "Crusher of Ores". Whenever it is nearby, all other machines will stop in their tracks. This is because

the machine carries he remains of the last living human. It killed that human by accident during a digging operation, and since then has been carrying the remains around and lamenting what it has done. Since it carries the remains, its laments somehow carry the weight of an order to the other machines, which halts their movement while the digger is around. After Frost becomes human, his first order is for the digging machine to stop this behaviour and start crushing ores again.