Mammals do not stop producing milk when they are pregnant. It is not uncommon for animals to give birth before the previous offspring is weaned. And therefore there are two babies who needs mothers milk. And the way mothers stop producing milk is that the baby stops suckling which stops milk from being produced.
So with cows what we do is impregnate them, then right after they give birth we take the calf away from them and start milking them. The biology of the udders do not make any difference if it is milked by a calf or a milking machine. So as long as it is being milked regularly it continues to produce milk.
The quality and quantity does get worse over time. So to get it back up again we impregnate the cow again. Then while the cow is pregnant we stop milking it so the cow stop producing milk for some time. After the cow gives birth again we take the calf away and start milking the cow with machines.
As for all the orphaned newborn calves most of them are culled. A lot of them are just killed and thrown in the compost heap or buried. Others are raised. Feeding them milk is expensive so they are weaned as soon as possible. It is possible to use milk replacement for very young calves which can be less expensive. This would be similar to feeding a baby the ground up baby food before it can eat proper solids. But it will take some milk before the calves can eat solids. The calves are then either slaughtered for their meat and skin or a few are raised to adult cows to replace older dairy cows.
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