Furthermore, the ordered sequence of primes that multiply to any given positive integer is unique. If 1 were considered prime, that wouldn't hold, as you could keep adding as many 1s as you like to the start of the sequence: (1,3,3,5), (1,1,3,3,5), etc. Thus 1 encodes no useful information here.
This is explained more formally here: https://en.wikipedia.org/wiki/Empty_product
When the sequence has one element, then that element is the product. So, 2 is the product of (2).
When the sequence is empty, the product is 1.