Why are written numbers universal across languages?

No, they’re not.

1. The reason East Asia uses the Decimal System with their own linguistic notations for the Numerals is because the Indian monks who were translating their texts probably decided that it would benefit the populace if the imagery had as little to do with extant Brahmi notations as possible.
2. The reason SE Asia uses the Decimal System with Pallavi or Vallabhi inspired notations because the most of their medieval rulers had administrations heavily influenced by Mainland Hindu Powers and often were married into Indian Royal Families, and thus found it easier to adopt the entire notational corpus.
3. The reason Europe and the Middle East use the Decimal Notation with this weird Kadamba-influenced Nagari notational base is because that’s what Brahmagupta used for some reason (probably because he was a migrant to Ujjaini & wasn't using his mother-tongue in his writings)- and his works were the ones being plagiarized all across the so-called “Islamic Golden Age”.
4. The reason most of Africa, Oceania, and the Americas use the Decimal Notation- again with this quirky Brahmagupta style- is because of Western Colonial Imperialism replacing their own local systems with this.

There’s absolutely no reason why Written Numerals ought to be the same for all languages. Such propositions are absurdities born from the work of Noam Chomsky, charlatan par excellence, whose absurd theories on the alleged universality of Language (which he appears to have pulled out of his arse because I've no clue how exactly he reached such conclusions and I haven’t read anything that even remotely comes close to justifying his ideas apart from ‘he’s a big-shot intellectual and you’re a stupid cook so shut up and listen’) have held back the development of Artificial Intelligence and Trainable Associative Algorithms for the better part of thirty years.

Languages - and by extension, Numerical notations- are completely Emergent Processes that are born of natural Civilizational complexity and thus, completely determinant on manipulation of Big Data. That’s why all major Linguistic Systems- except Chinese and Japanese- are Mesopotamian in origin. That’s why the primary Numerical notation is Indian in origin. That’s why Dan Everett is a hero for his work on the Piraha.

As for why the various Indian Numerical Systems, be they of Pallava, Vallabha, or Nagara origin, look the same- it’s because the Emperor Samudradeva conquered the Continent and established an Empire that would last half a millennium