Also in practice we work with number representations, not number themselves. So there are some patterns where the representation is influenced by which base we encode them into. That's not something specific to primes of course.

For example, length in term of digits or equivalently weight in bits will carry depending on the base, or more generally which encoding system is retained. Most encoding though require to specifically also transmit the convention at some point. Primes on the other hand, are supposedly already accessible from anywhere in the universe. Probably that's part of what make them so fascinating.