Is it true that if pointed spaces $X, Y$ have the same homotopy groups $\pi_n(X) \cong \pi_n(Y)$, then they have the same stable homotopy groups $\pi^S_n(X) \cong \pi ^S_n(Y)$?
In particular, is this true for $\mathbb{R}P^2 \times S^3$ and $S^2 \times \mathbb{R}P^3$ (whose homotopy groups are obviously isomorphic)?
No, a counterexample is the rational sphere $S^{2n}_\mathbb{Q}$ and $K(\mathbb{Q},2n) \times K(\mathbb{Q},4n-1)$. By the work of Serre these have the same homotopy groups, though it is easy to see they are not homotopy equivalent. The stable homotopy of the former is easily seen to be $\mathbb{Q}$ concentrated in degree $2n$ while the stable homotopy of the latter is $\mathbb{Q}$ in degrees $2n,4n-1,6n-1$ since (i) products split stably as $X \vee Y \vee (X \wedge Y)$ and (ii) the Moore spectrum for $\mathbb{Q}$ is equivalent to the Eilenberg-MacLane spectrum for $\mathbb{Q}$.
