For $\mathbb{R} P^{n}$ we have,
$$ H^*( \mathbb{R} P^2 ; \mathbb{Z}_2) \cong \mathbb{Z}_2 [ \alpha ] / (\alpha^3)$$
where $|\alpha|=1$.
Let us assume there exists $X$ such that,
$$ H^*( X ; \mathbb{Z}_2) \cong \mathbb{Z}_2 [ \beta ] / (\beta^3) $$
but $| \beta |=2$.
Then the rings have the form, $a_0 + a_1 \alpha + a_2 \alpha^2$ and $b_0 + b_1 \beta + b_2 \beta^2$.
So as polynomials these rings look the same except for the condition given on the variables $\alpha$ and $\beta$.
How does the dimension of $\alpha$ and $\beta$ come into play when viewing the polynomial rings?
I think, naively, I would assume these space are isomorphic and that dimension doesn't add any strong information about the structure of the spaces.