Question: As onomatopoeia, the words used for animal sounds are often quite similar across many languages. However, there are non-trivial differences, even for something as common as the croak of a frog.

I was wondering if these differences could generally be grouped (or explained) according to language family or if it's basically just chance (eg, the term that caught on due to some local story or song).

Another way to put this is to ask to what extent the animals actually sound different to speakers of different languages (without getting too philosophical).

Answer: 
Onomatopoeia is non-arbitrary, but that doesn't mean it's immune to the normal processes that happen to any arbitrary word—including:

* arbitrary historical choices of onomatopoeia (like @acattle commented about "ribbit" above; copy-pasting):

> 
> Incidentally, frogs make a wide array of sounds. Only a single species actually goes "ribbit", the Pacific Treefrog indigenous to the US west coast. Despite this, "ribbit" is the frog onomatopoeia of choice for most English speakers around the world. The reason is because [early Hollywood movies used the frog's distinctive call as a stock sound effect](http://www.frog-shots.com/Who_says_RIBBIT_.html). This explains why English's frog onomatopoeia is [noticeably different from other languages'.](http://en.wikipedia.org/wiki/Cross-linguistic_onomatopoeias#Frog_croaking)
> 
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* inheritance through language families, like any other cognate

There looks to be a pan-Slavic *[h]am [h]am* onomatopoeia for "eating food", for example (English *nom nom* or *yum yum*), which doesn't appear elsewhere in the Wikipedia list (although Turkish *ham hum* is close.)

* compliance with language-specific phonotactics

Turkish takes \**mjau* for "meow", and ends up with *miyav*. A lot of languages don't have initial /mj/ clusters; Turkish seems to make that official even for meowing. (Turkish doesn't have initial /m/ at all natively, admittedly; but /mj/ is a bigger hurdle than just /m/.) The final -av is another concession to Turkish phonotactics.

* areal diffusion

The Modern Greek sneeze, for example, /apsu/, is pretty obviously borrowed from the Turkish (and more onomatopoeic) /hapʃu/; dropping /h/ and /ʃ/ > /s/ is just fitting /hapʃu/ to Modern Greek phonotactics. I have heard, to my profound shock, my cousin-nephew in Athens spontaneously utter /auts/ "ouch", when the Greek exclamations I knew were /ax, ox/. The fact that /twit/ for bird noises is shared between English and Dutch, but not German, points to either common heritage, or English influence on Dutch via comics (which I suspect for Finnish *tviit* and Tagalog *twit*).

I only realised the Greek/Turkish link of /apsu/ last year, and I was shocked, because I assumed onomatopoeia was completely non-arbitrary. But there you have it. Onomatopoeia does get culturally conventionalised, and is subject to the same borrowing and diffusion as any meme, let alone any lexeme. Just like kids drawing houses with chimneys, even if they've never seen a chimney. 

And not that I'm equipped to answer your final question, but if you pressed a bilingual in Greek and English like me, I would not be able to tell you that Greek [ɲau] is more or less authentic a realisation of a cat's vocalisation than the [mjau] that occurs just about everywhere else. (Latvian in any case is happy to use both: *mjau, ņau*.)
When I married my husband, from Cuba, I was SHOCKED to find out that the dogs there spoke Spanish! They didn't say, "woof-woof" or "bow-wow," but "jao-jao" (translated: how-how!) That's actually closer than bow-wow! I don't know what other animals say. I just laughed so hard at that realization, that I never went any further.