A product can include just itself... sort of. I know it's not intuitive, but it's there. The formal definition is usually expressed as a product of powers of primes and in simplified form is n = p
1^i * p
2^j ... and so on. N is the number you want to make out of primes, and p are the primes. E.g. 6 = 2^1 * 3^1. Some numbers just have a single p like 8 = 2^3. Prime numbers are just the simplest case of single p and the power it is raised to is just 1. E.g. 5 = 5^1.
And to get a bit more abstract, technically this means that all numbers include all primes as part of their product using this definition since p^0 = 1. I.e. the full product of primes for 5 is: 5 = 2^0 * 3^0 * 5^1 * 7^0 * 11^0 * ... etc. This is allowed since it still uniquely defines the number despite having an infinite number of terms. E.g. you will always have 3^0 and 5^0 as factors in 14 and changing those or any other term with power 0 will change the number. The number 1 breaks this, because 1 to any power is still 1 so it doesn't change the result. This means that if p were allowed to be 1 there would be no more unique factorization where n = p
1^i * p
2^j ... and so on. E.g. 5 = 1^0 * 5^1 = 1^1 * 5^1 = etc.
Edited: great question by the way! That's exactly the right kind of thinking to approach the problem.
