Do languages other than English have a version of what in English is called Pig Latin?
“Igpay Atinlay.”
No, no, Uvula Donor, that should be “Atinalay Orcorumpay” 
And it doesn’t directly address your question, but this Staff Report might be of tangential interest.
They’re called Ludlings and they are common in languages from all over the world. I’m looking at The Handbook of Phonological Theory (edited by John Goldsmith) right now and the “Language Games and Related Areas” chapter by Bruce Bagemihl formally describes ludlings in Tigrinya, Amharic, Tagalog, Cuna, Javanese, Hanunoo, Thai, Mandarin, French, Fula, Chasu, Bakwiri, Sanga, and a few others.
So, yeah, they are pretty common.
Not really the same thing at all, but the French have ‘verlan’, a form of speech where new slang is created by saying it backwards (verlan is ‘l’envers’ - meaning backwards - said backwards :p). So only certain words are changed, and become kind of a young people’s patter - the only one I can think of is ‘meuf’ = femme = woman - I’m sure clairobscur will come along with some more.
Do you mean featuring the same mechanism - moving the first consonant sound to the end of the word and adding “-ay” - or just the general feature of modifying a language to create a “code” of sorts? Off the top of my head, similar things certainly exist in Spanish and French, though I can’t remember their names. In Argentina there’s a code that switches around syllables, so that “vidrio” (glass) becomes “driovi”.
I suspect these are universal. Sorta tangentially related is the phenomenon in Dyirbal, one of the Australian Aboriginal languages, called “mother-in-law language”. I think it exists in some other Aboriginal languages as well; there’s a cultural taboo around speaking within the earshot of certain in-laws of the opposite sex. So an entirely new lexicon exists - that is, the same language (in terms of grammar and sounds) is used, but with completely different words. Thus people can talk without violating the taboo. As the Aboriginal languages have faded, this has disappeared; I don’t think it’s been in regular use since the '30s or so.