I think a lot of people consider uniqueness of prime factorization to be obvious just because it's stated as true so frequently.

If you want to convince someone it's not obvious, I would ask them a simpler, weaker question:

Can you find four different prime numbers, a b c d, so that ab = cd? If not, why not? You may not cite unique-prime-factorization.

They will find themselves really wanting to cite unique prime factorization, and unable to prove it otherwise, which will convince people that it's not obvious.