Let $X$ be a random vector with probability density $p$.
In the scalar case I have learned that if the characteristic function of $X$ is real analytic, then all moments exist and $p$ is determined by the moments.
Is this also true in the multivariate case?
Yes. The characteristic function $\phi(s) = {\mathbb E}[\exp(i s\cdot X)]$ is the complex conjugate of the Fourier transform of the density, and Fourier transform is one-to-one on $L^1$. If it's real analytic, $\phi(s)$ is determined by the coefficients of its series expansion at $s=0$, which are the moments.