There is no single word for this concept that is in common use; usually, the terms people use for the types of animal with which humans have the greatest involvement are fairly species-specific, as you have already implied.
However, the simple description 'young animal(s)' covers the idea quite well for the young of any animal that isn't conventionally thought of as a bug or a creepy-crawly.
Take a look at this Ngram (which I have set to display the maximum possible smoothing of 50) to compare the prevalence of some terms (most of which will be very familiar) for young animals in books published between 1800 and 2000. (I have excluded terms like 'kit' and 'fry', which would generate too many false positives because of their overlap with senses of the word that have nothing to do with animals.)
You will notice that youngling, which is very much a dialect term, always lies at the very bottom of the frequency distribution.