For any Dirichlet series, $$f(s)=\sum_{n=1}^\infty \frac{a_n}{n^s}$$ is the sequence, $a_n$, always unique to $f(s)$? In other words, is it possible to show that $a_n$ is the only sequence that will ever satisfy the series being equal to $f(s)$?
If this is not true, could someone try to provide a counter example if possible?
If we assume that your series converges absolutely for all large $s$, then, yes, expansion of $f(s)$ as Dirichlet series is unique. See Theorem 4.8 at http://www.math.illinois.edu/~ajh/ant/main4.pdf for proof.
Otherwise, you should be able to provide a counterexample.