Expanding on Stopple's comment above, I believe the following argument based on Landau's explicit formula answer's the question.

Here is a generalization of Landau's explicit formula for the zeros of the Riemann zeta-function which is exercise 8.4.8 in M. Ram Murty's book Problems in Analytic Number Theory: Let $F$ be in the Selberg class, $n>1$ be a positive integer, and $T>1$. Then $$ \sum_{|\gamma|\leq T} n^\rho = -\frac{T}{\pi}\Lambda_F(n) + O( n^{3/2}\log T )$$ where $\rho=\beta+i\gamma$, $\beta>0$, runs over the non-trivial zeros of $F(s)$. Here the coefficients $\Lambda_F(n)$ are defined by $$ -\frac{F'}{F}(s) = \sum_{n=1} \frac{\Lambda_F(n)}{n^s}.$$

Now suppose that $F$ and $G$ are in the Selberg class and have the same zeros (with multiplicity). Then we deduce from Landau's formula that $$ |\Lambda_F(n) - \Lambda_G(n)| \ll \frac{n^{3/2}\log T}{T} $$ for all $n>1$. Letting $T\rightarrow \infty$, it follows that $\Lambda_F(n) = \Lambda_G(n)$ for all $n>1$. This implies that $F=G$.

Expanding on Stopple's comment above, I believe the following argument based on Landau's explicit formula answers the question.

Here is a generalization of Landau's explicit formula for the zeros of the Riemann zeta-function which is exercise 8.4.8 in M. Ram Murty's book Problems in Analytic Number Theory: Let $F$ be in the Selberg class, $n>1$ be a positive integer, and $T>1$. Then $$ \sum_{|\gamma|\leq T} n^\rho = -\frac{T}{\pi}\Lambda_F(n) + O( n^{3/2}\log T )$$ where $\rho=\beta+i\gamma$, $\beta>0$, runs over the non-trivial zeros of $F(s)$. Here the coefficients $\Lambda_F(n)$ are defined by $$ -\frac{F'}{F}(s) = \sum_{n=1} \frac{\Lambda_F(n)}{n^s}.$$

Now suppose that $F$ and $G$ are in the Selberg class and have the same zeros (with multiplicity). Then we deduce from Landau's formula that $$ |\Lambda_F(n) - \Lambda_G(n)| \ll \frac{n^{3/2}\log T}{T} $$ for all $n>1$. Letting $T\rightarrow \infty$, it follows that $\Lambda_F(n) = \Lambda_G(n)$ for all $n>1$. This implies that $F=G$.