I’m going to guess, based on no evidence or research whatsoever, that English-speakers were calling youthful goats “kids” centuries before we applied the same term (colloquially) to the young of H sapiens. I’m also going to guess that the colloquialism for (human) children started c. 1750, again, based on no evidence or research.
“Kid” as “baby goat” dates from c. 1200.
“Kid” as “child” dates from c. 1590 but didn’t fully catch on until the 1840s.
From the OED, there are also several unrelated words spelled and pronounced the same: (1) a bundle of twigs for burning or embedding in a bank to give firmness to loose soil (unknown origin, first appeared c.1350); (2) a seed-pod of a leguminous plant (unknown origin, first appeared c.1722); (3) a small wooden tub for domestic use (unknown origin; first appeared 1769); (4) nonsense, often in the form “no kid”, meaning “I am not kidding” (from the verb, first appeared 1874); (5) knowledge or a place that is known or familiar (related to couth, first appeared before 1400). There are also six more senses used as a verb.
let me add that in German, a young goat is also referred to as a Kitz (phonetically +/- identical to kids)
Only that in this meaning, Kitz in German is not limited to young goats; it’s applied to the offspring of a whole range of mammals. I think it’s most frequently applied to young deer; for young goats, alternative words, such as Zicklein, exist and are, at least in my perception, more common.
I’ve seen that “hedum” in Latin is translated to “kid” and also referred to children. Julius Caesar made a pun on the connection.
The German “Kind”, for (human) child, might also have been an influence (is that etymologically related to “Kitz”?).
Not according to Wiktionary and an online etymology dictionary that I checked. Kind is said to go back to an Indo-European root for “giving life” (which is possibly also behind Latin genus and words derived from that, such as “generation”), whereas Kitz is apparently based on an onomatopoeic call.
There’s a whole mess of words that may or may not be related and perhaps have influences on each other: kid, goat, kit, kitten, kitling, chit, child. The etymologies are unclear and would take a specialist to have any certainty, if it’s even possible.
Etymology is not such a vague discipline. Normally, the way words change over time is not random but follows an overall pattern, such as the Great Vowel Shift in the Germanic languages. This allows linguists to retcon what the root of a word must have been long ago. We don’t have any written record of Indo-European, for instance, but many Indo-European roots have been reconstructed with a good amount of certainty.