Sorry for strangely worded title. The intended meaning is the generating functions which are not divisible by other generating functions, not functions for generating prime numbers.
With this out of the way, are such functions studied? Are they called "prime". My understanding is that division isn't defined for all generating functions, so are there any parallels between these and prime numbers? If they are, what field of mathematics studies it?
They're called "generating functions with zero constant term and nonzero $x^1$ coefficients".
In the ring of formal power series over a field, the irreducible elements are exactly those where $a_0=0$ and $a_1\ne 0$.
If $a_0\ne 0$, the series is invertible, and so not usually considered irreducible.
On the other hand, if $a_0=a_1=0$, you can divide by $X$ to get a nonunit quotient -- so those series are reducible.
But series with $a_0=0$ and $a_1\ne 0$ are not themselves invertible, and can only factor if the constant term of one of the factors is nonzero (and therefore the factor is a unit).
And yes, these irreducible elements are the same as the prime elements because the ring of formal power series is a unique factorization domain. An element $p$ of a ring is called "prime" if whenever $p$ divides a product it divides at least one of the factors. In unique factorization domains, this is equivalent to being irreducible.
On the other hand, "prime" is not a very interesting property in this setting because it turns out that all primes in the ring are associate -- that is, for every two primes one will equal some invertible series times the other, and the "prime factorization" of every nonzero power series is just $x^n f(x)$ where $f$ is some invertible series.
These things generally belong to ring theory, also known (slightly more precisely) as commutative algebra.