Does FACTORING have optimal substructure or analog to it?
No.  There's no known optimal substructure like that.  Such a substructure would naturally lead to algorithms for factoring large composite numbers, which is a notoriously hard problem.  For numbers that are products of two primes, there is no known optimal substructure; these are the kinds of numbers that are used in the RSA cryptosystem, and factoring them has been studied in great depth in the literature.  However, the best factoring algorithms known don't use optimal substructure; instead, they use other number-theoretic ideas.

Another way to put it: recursion does not seem like a useful approach to factoring products of two primes efficiently.  The natural way to recurse is to find a divisor $d$ of $n$, then recurse on $d$ and $n/d$; but that's a chicken-and-egg problem, because finding a divisor of $n$ is exactly the problem that's hard.  For instance, when $n$ is a product of two primes, factoring is the problem of finding a (non-trivial) divisor of $n$.  So, the problem is that, to figure out what to recurse on, you first need to solve the problem itself.  My thanks to @Raphael for this helpful way of thinking about things.