> In comparison the fundamental theory of arithmetic is also strange but there is a good intuition behind it: any non prime number is a product of primes, because if it wasn't it would be prime itself!

That's true, but it doesn't prove the fundamental theorem of arithmetic. You need to show that e.g. if p,q,r,s are four different primes, then pq does not equal rs.

(The important part of the theorem is that the prime factorization of an integer is unique.)