Maybe not all cultures per-se, but I'm quite the movie nut, I've seen it in Asian movies, obviously western movies and even bollywood movies, they all seem to signify similar things to with similar farewell connotations. How did this come to be?
For one thing, it's an old song. In the 1700's when it was formally written and submitted to records in Scotland, it was words for an already old well known folk tune.
"The following song, an old song, of the olden times, and which has never been in print, nor even in manuscript until I took it down from an old man. - Scottish poet Robert Burns. 1788
Musically, its pentatonic scale is very compatible with Eastern Asian music already. Take this national song in Thailand for example, Samakkhi Chumnum: https://youtu.be/sNYRPG5MkPU?t=3m26s . The lyrics are about their King and unity, so it's significance and meaning differs greatly from Auld Lang Syne. In japan, the song is completely different lyrics (Hotaru no Hikari, about student determination), but is indeed used in a similar fashion to Auld Lang Syne...music being a very potent cultural pillar, to celebrate graduations and as a weird quirk, being used for formal business closings.
This phenomena is called contrafactum, the complete substitution of text within a piece of music. This is very common for many songs, especially popular lullabies, nursery rhymes, and folk tunes. If a song is catchy, people tend to change it to their own preferred lyrics and language.
Take "Twinkle, twinkle Little star", which is originally a French folk song (and made popular by Mozart) called 'Ah! vous dirai-je, maman", but no shocker, the tune is identical to baa-baa-black sheep, the alphabet song, and many others.
This same can be said for national anthems, Estonia and Finland have the same tune for example.
Sometimes this contrafactum is intentional. During the US Civil war both sides parodied each others songs, from Yankee Doodle (itself taken from a folk song) to John Brown's Body. Most of the time lyrics aren't changed for propaganda, or other intentional purpose -people just write new words and they're catchy enough to spread. edit-typo
It's also used in quite a number of Japanese shops to denote closing time. I'd like to append a question as to why this is?
Koreans actually used the tune of Auld Lang Syne as the melody for the national anthem for the provisional government of Korea when Korea was still ruled by Japan. This is a scene from a Korean TV drama depicting the guerrilla in Manchuria during the Japanese rule