I realise that one could define two spaces to be "homology equivalent" if they have the same homology, as people say, i.e., if they have all homology groups isomorphic. But I find this unsatisfactory because this is not how it goes for homotopy. Homotopy equivalence is not defined as the equality of all the homotopy groups, but as the existence of continuous maps to and fro that compose to maps homotopic to identity maps. And we know that this is strictly stronger than the equality of all the homotopy groups together.
Is there a similar, natural notion of spaces being homologous, which then turns out to be at least as strong as all the homology groups being isomorphic?
Yeah, such notion would not be useful, except for the special case when all homology groups are trivial.
That's true, but there is a notion of weak homotopy equivalence, which is a continuous map $f:X\to Y$ such that $\pi_n(f):\pi_n(X)\to\pi_n(Y)$ is an isomorphism for any $n$. This is strictly stronger than requiring $\pi_n(X)$ to be isomorphic to $\pi_n(Y)$, we want the isomorphism to arise from a continuous map.
This idea already is quite useful, and in fact Whitehead showed that every weak homotopy equivalence between (homotopy) CW complexes is actually a strong homotopy equivalence. For more information on weak homotopy equivalences see the wiki entry.
So, by analogy we could define a "homology equivalence" as a continuous map $f:X\to Y$ such that $H_n(f):H_n(X)\to H_n(Y)$ is an isomorphism for any $n$. Although I don't think such maps are widely studied, they don't behave as well as weak homotopy equivalences. One thing to note is that by combining Whitehead's theorem with Hurewicz theorem we get that a homology equivalence between simply connected CW complexes must be a strong homotopy equivalence. Although without "simply connected" assumption this is not true for homology, see here.