I don't know if it's possible to obtain further information on the ring structure in
general. However, if $A$ is the mod-p cohomology ring of a finite group, a result
of Benson and Carlson states that if $A$ is Cohen-Macaulay, then $P(t)$ satisfies
the functional equation
$$P(1/t) = (-1)^d P(t)\hspace{90pt}(\ast)$$
where $d$ is the Krull dimension of $A$. Conversely, if a given $P(t)$ doesn't satisfy
this equation, you know that $A$ isn't Cohen-Macaulay, i.e. the depth of $A$ is less
than its Krull dimension.

In case $A$ has Krull dimension 2, than $A$ is Cohen-Macaulay
iff $P(t)$ satisfies $(\ast)$. These results can be found in the paper Benson, Carlson: Functional equations for Poincaré series in group cohomology. Bull. London Math. Society 26(1994), 438-448.

As an example for groups with identical Poincaré series you can take $E := \mathbb{Z}/2 \times \mathbb{Z}/2$ and $D_8$, the dihedral group of order 8. Their cohomology rings
$$H^\ast(E;\mathbb{F}_2)=\mathbb{F}_2[x,y],\;\; |x|=|y|=1$$
$$H^\ast(D_8;\mathbb{F}_2)=\mathbb{F}_2[x,y,z]/(xy),\;\; |x|=|y|=1, |z|=2$$
aren't isomorphic (since only one is a domain) but both have Poincaré series
$P(t)=\frac{1}{(1-t)^2}$.

**Added:** There is a paper of R. Stanley (who is also active on MO) that contains some properties of the Poincaré series that may be of interest. I just quote a few:

If $A$ is Gorenstein of Krull dimension $d$, then $P(1/t)=(-1)^dt^aP(t)$ for some integer $a$. This generalizes $(\ast)$ (with a=0) because the mod-p cohomology ring of a finite group is Cohen-Macaulay iff its Gorenstein.

$A$ is a complete intersection with generators of degree 1 iff $P(t)$ has the form
$$P(t)=\frac{\prod_{i=1}^l(1+t+\cdots+t^{m_i})}{(1-t)^d}$$

A sequence of homogeneous elements $x_1,...,x_k \in A$ of positive degrees $n_i$ is regular iff $P(A,t) = P(B,t)/\prod_i (1-t^{n_i})$ where $B=A/(x_1,...,x_n)$.