Definitely not just a US thing, and it really is just the difference in syllables.
There's a fancy name for it in linguistics, but shortening words in speech and writing is a thing that's human.
Not every language shortens every word, obviously. But when something is common enough in usage, it gets as short as possible.
Zero is one of two single digits with multiple syllables. I'm still surprised seven hasn't gotten trimmed down in speech yet. Closest I've heard is "seb'n", which is a teeny bit faster because of three way the lips move with a b vs a v.
Since there's a letter sound that closely matches the written numeral (and, being real, they're written the same, just not machine printed the same), it just makes sense that "oh" gets substituted rather than it getting elided in some way.