Prime numbers are a pattern; take the natural numbers - starting after 2, exclude every number that isn't 2, starting after 3, exclude every number that isn't 3, etc.

It repeats like this predictably. Even though it changes, the way in which it changes is also predictable. Their repetition and predictability make prime numbers a pattern.

Out of the fundamental pattern of prime numbers, higher-level patterns also appear, and studying these patterns is a whole branch of math. You can find all kinds of visualizations of these patterns, including ones linked in this thread.

It's not that you're seeing a pattern that's not there, it's that you're seeing a pattern that gradually becomes infinitely complex.