Wikipedia -- the first and last authority on such things -- carefully writes in an entire sentence unto itself, "The number 1 is by definition not a prime number," suggesting just how arbitrary this is.

A measure of the arbitrariness is the history, which is that 1 was considered prime up to the 19th century and was a matter of fashion during the 19th century. That suggests that unique factorization is not, in itself, enough to motivate the definition. Perhaps its extension to the gaussian integers or the more radical version for general number rings prompted the definiton.