I’m sure that people all over the world sneeze and cough the same way, because they’re involuntary, but how about laffing? Does everybody do “ha ha”, or is that something cultural or language-specific?
In Latin American spanish it’s the same sound but spelled “jajaja” or “jejeje”.
The “words” used for laughter, like all other onomatopoeia, are cultural.
There are a lot of Japanese onomatopoeia for laughter. A lot of the variation is tied up in traditional cultural expectation of different ages and sexes; for instance, women are expected to be more modest than men, so their laughs are less boisterous and more closed-mouthed (and supposed to be executed behind a hand hiding the mouth). There are also variations related to loud laughter, snickering, chuckling, and the like. Also the perceived vulgarity of the subject of the laughter.
Japanese is complicated.
ETA: However, something sounding like “Hahaha” (“はははは”) is the “default” laughter for the “default” person: an adult male laughing “normally”.
:What fascinates me is that cows don’t go “moo”, cats go “meow” and dogs don’t go “woof woof” in other languages. 
And giggling is jijiji and that fat guy in red laughs jojojo.
In Italian it is usually vowel+H
Most people sneezing within earshot of my office do some recognizable variation on “a-choo”, but a former Pakistani native co-worker was the only person I’ve ever heard whose sneezes sounded like “heh-shoo!”
My Italian teacher related that the Italian verbalization for a sneeze, instead of “ah-choo,” was supposed to be azzìn. But that no Italians he’d ever known in real life sneezed by going “azzìn.” Except his mom. If he heard “azzìn” in a crowd, in church, or anywhere, he’d know it was his mom.
The Thai word for 5 is pronounced “ha.” So where an English speaker might end a Facebook post with “hahaha” or “lol,” a Thai will close with “555.”
I get the impression that the OP was asking about how people actually laugh, not how they write the representation of laughing (otherwise the part about sneezing and coughing being involuntary would be really incoherent and incongruous). So a possibly better phrasing might be, ‘do humans all over the world make a very similar sound when they laugh and do it in a highly similar way?’
Assuming that’s what was meant, it’s an interesting question. I’d be curious to know for sure, because although my guess would be yes, I have no knowledge to confirm that.
Portuguese speakers on the Internet will often type out “kkkkkk” for laughing. The letter K in Portuguese is pronounced “ka,” so “ka ka ka ka ka.” Kinda the same sound.
My mother in law doesn’t sneeze like anyone else. It sounds like she’s screaming, loudly. AHHHHHHHHHHHH!. My husband says he used to try to teach her how to sneeze properly, "you say “ah-choo, mom. with the ‘choo’.” Now if she remembers, it’s “AHHHHHHHHHHHHHHHHHH-choo”. 
The King of Korean laughs Lee Soo Geun:
https://www.youtube.com/watch?v=fPyp0_-EuIE
Thai’s write 555 to mean laughing. You guessed it, the thai for 5 is “ha”.
Mary Poppins explains it as follows with Bert’s help: I Love To Laugh - Mary Poppins - YouTube (skip to 1:00 mark)
So I looked it up, and I find the Italian for “achoo” is eccì. Which is close enough to what I remembered; either I misremembered (it was many decades ago; I was 12) or my teacher was talking dialect.
For future reference, you can link to a specific time on YouTube here.