When did "kid" start to mean "child"?

Solution 1:

Kid as young goat is from the 1200s with

Extended meaning of "child" first recorded as slang 1590s, established in informal usage by 1840s.

Dictionary.com defines kid as (informal) child.

You would use it in direct conversation with persons you know well

"Does your kid collect stamps?"

although I don't see it as too informal to ask someone, "Any kids?" instead of, "Any children?"

Solution 2:

"Kid" is definitely less formal than "child", although the proportion of people for whom it is offensive is probably small now.

As to the timing, I have no information.

Solution 3:

According to Etymology Online, the use of "kid" to refer to a human child was established in informal English usage by 1812, but was used as slang (not sure what the difference is in connotation here between "informal" and "slang") as early as 1590.

The term may have first been applied to human children in reference to similarities between a goat kid's cry and a newborn's. Or, simply, as reference to the child as young, similar to use of the term "pup" (the name for many mammals' young, mainly canine but also otters and other aquatic mammals)