It looks the way it does because we like to see patterns even where there are none. E.g. you see a number 696969 and this seems more significant than 482649 for whatever reason
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.
These patterns were documented and well understood starting in BC times by Erasthosenes and learning them as part of prime number theory is a 101 course in tertiary maths education. So it's really really weird for anyone to say "there's no patterns". There are and they are extremely well understood and known.
Here's a simple pattern; All prime numbers above 2 are odd. Well duh right? Otherwise they'd be a multiple of 2, not prime.
Well let's extend this. All prime numbers above 6 are of the form 6n + 1 or 6n +5. Otherwise they'd be a multiple of 2 or 3.
Once more; All prime numbers above 30 are of the form 30n + one of [1,7,11,13,17,19,23,29]. Anything else would be a multiple of 2,3 or 5. You can extend this forever. Note each time we do this we're reducing how many numbers could possibly be prime. From 1/2 to 2/6 to 8/30 numbers possibly being prime. Keep going with this and you'll converge to the prime counting function.
Basically whenever you have a composite number there's well understood periodic gaps in primality. People understand this more intuitively for base 10 where anything ending in 0,2,4,6,8 is a multiple of 2 and anything ending in 0,5 is a multiple of 5 hence you only get primes ending in 1,3,7,9 when writing in base 10 but this idea works for any composite number. This leads to the extremely well known and well understood patterns you get when you graph primes in various ways.