Wikipedia “In number theory, the fundamental theorem of arithmetic ...the unique-prime-factorization theorem, states that every integer greater than 1 either is prime itself or is the product of prime numbers”...The theorem is stating...first, that (a composite number) can be represented as a product of primes...
Example A: 6 = 2 x 3 I take it that “is the product of” means that if we carry out the calculation and multiply these primefactors only (2 x 3) we will get the answer 6. This makes sense.
Example B: 12 = 2 x 2 x 3 I am assuming that in math we can only calculate one set of numbers at a time. So we can either do 2 x 2= 4 then 4 x 3 =12. or 2 x 3= 6 then 2 x 6 =12
4 and 6 are not prime numbers so how can we say that the composite number 12 is the product of prime numbers? Isn’t it the product of one prime number and one composite number? 12= (4 x 3) or (2 x 6).
So any non prime number that has more than 2 prime factors cannot be the product of primes because by multiplying 3 or more prime numbers you will get a composite number before you get to the answer. Example. 20= 2 x 2 x 5 20= 4 x 5 or 20 = 2 x 10 To solve this problem you will either get a 4 or a 10 before you can multiply it by the last prime number in the equation.
The Wikipedia article assumes that "a product of ... numbers" means that you have two or more numbers and that you will multiply them all together. For example, in that sense, $12=2\times2\times 3$ is a product of prime numbers.
If they wanted to say what you think they said, they would say something like "every number ... or is a product of exactly two prime numbers". They would then be wrong, as you have already noticed.