Can we classify all finite CW complexes $X$ such that for each $i$ there is some isomorphism $\pi_i(X) \rightarrow H_i(X)$? Note that it is not hard to classify all complexes for which each isomorphism is given by the Hurewicz map.
I also don't see a motivation for this, but anyway: if $i=0$ is allowed, there is not a single example ($\pi_0$ is always finite, $H_0$ never. For the empty space, $\pi_0$ is empty and $H_0$ is not). Apart from $i=0$, the only connected finite CW-complexes with only finitely many (edit: abelian) homotopy groups are tori. (J.-P. Serre, Cohomologie modulo 2 des complexes d’Eilenberg-MacLane, Comment. Math. Helv. 27 (1953), 198-232.)