Are there any set of axioms that completely characterize the Riemann zeta function? i.e. like Ressayre axioms for the exponential function in an exponential field or functional equations.

Perhaps you are looking for something like Hamburger's Theorem? It states, essentially, that the only Dirichlet series with a finite number of singularities satisfying the same functional equation as the zetafunction is the zetafunction. You can find the details in Titchmarsh's book. Googling I found the following link: http://www.mat.univie.ac.at/~esiprpr/Zetaproc/patterson.pdf 


I don't know if this is in the spirit you're looking for, but there is the Selberg class  an attempt at axiomatizing $L$functions, requiring a Dirichlet series, functional equation of a certain type, analyticity, and an Euler product (typically)  and it would be possible to impose extra conditions to isolate the zeta function. In particular, all degree 1 elements are known to come from Dirichlet Lfunctions (this was proved by Kaczorowski and Perelli, and then reproved by Soundararajan). Thus, requiring the degree and the conductor to both be 1 should isolate the zeta function. 


Hamburger's Theorem (see Titchmarsh 'Theory of the Riemann Zeta Function' $\S$ 2.13) is in some sense an axiomatic characterization of $\zeta(s)$ among all Dirichlet series by its functional equation. It says: Let $f(s)=\sum_n a_n n^{s}$ a Dirichlet series absolutely convergent for $\sigma>1$ such that for some polynomial $P(s)$, $G(s)=P(s)f(s)$ is an integral function of finite order. Suppose $$ f(s)\Gamma(s/2)\pi^{s/2}=g(1s)\Gamma((1s)/2)\pi^{(1s)/2} $$ where $g(1s)=\sum_n b_n n^{1s}$ is absolutely convergent for $\sigma<\alpha<0$. Then $$ f(s)=C\zeta(s) $$ for some constant $C$. Hamburger's theorem was a motivation for Hecke's study of Dirichlet series with functional equations generally, leading to his work on automorphic forms. 


There are in fact at least two axiomatic characterizations of $\zeta(s).$ One of them is given by Hecke and one of them was given by Hamburger. Hamburger's Theorem states: Suppose $D(s)$ is a dirichlet series, convergent in some half plane and whose coefficients have polynomial growth. If $$$$ 1) There exists a polynomial $P(s)$ so that $P(s)D(s)$ is entire and of finite order. 2) $D(s/2)$ is also dirichlet series. That is, the coefficients of $D(s)$ are supported on squares. 3) If $\xi(s)= \pi^{s}\Gamma(s)D(s)$ then $\xi(1/2s)=\xi(s).$ Then $D(s)=C\zeta(2s).$ Hecke's version states: Suppose $D(s)$ is a dirichlet series, convergent in some half plane and whose coefficients have polynomial growth. If $$$$ 1) $(s1/2)D(s)$ is entire and of finite order. 2) The coefficients of $D(s)$ have arbetrary support. 3) If $\xi(s)= \pi^{s}\Gamma(s)D(s)$ then $\xi(1/2s)=\xi(s).$ Then $D(s)=C\zeta(2s).$ A little extra information: These theorems cannot be combined. The socalled "Big Mac Theorem" where$$$$ 1) There exists a polynomial $P(s)$ so that $P(s)D(s)$ is entire and of finite order. 2) The coefficients of $D(s)$ have arbetrary support. 3) If $\xi(s)= \pi^{s}\Gamma(s)D(s)$ then $\xi(1/2s)=\xi(s).$ produces infinitely many linearly independent dirichlet series! Sources: 

