What is the motivation for requiring that the square of a differential be $0$ for a complex, aside from enabling us to speak of the homology of a complex?
Other homological notions like chain maps, homotopic maps, homotopy equivalences seem to be meaningful without any restriction on the differential (of course, no longer do homotopic maps induce isomorphisms on Homology, for Homology no longer is meaningful).
If we ignore any connections to Homology, is there some other moral reason to want that the differential square to $0$?
The main motivation for wanting $d \circ d = 0$ is because it arises naturally in so many applications. As others have already said in comments, exact sequences are common and important, and you often get chain complexes by applying functors to exact sequences. At that point, homology measures how far the new sequence is from being exact: often a very interesting question.
More generally, people certainly study diagrams of the form $$ X_1 \to X_2 \to X_3 \to \cdots $$ with no condition on the maps, and then you can examine maps between such diagrams — the analog of chain maps. This happens all the time in algebra; for example, $X_i$ might be a submodule of $X_{i+1}$ for each $i$, giving a filtration of $\bigcup X_i$.
In the case when $d^2 \neq 0$, although I suppose the analog of "chain homotopy" could be defined, it is not clear what use it would be. What can you deduce if $f$ and $g$ are "chain homotopic" in this more general sense? What does it tell you in the case where each $X_i \to X_{i+1}$ is injective? Is surjective? When considering honest chain complexes, the definition of chain homotopy is motivated by the definition of homotopy in topology, and indeed, homotopic maps between topological spaces induce chain homotopic maps on their singular chain complexes: there is a mechanism for producing chain homotopies, at least in this one situation. Is there any mechanism that produces this analog of a chain homotopy? Without good motivating examples and/or interesting consequences, it doesn't seem worthwhile (to me, at least) to devote too much energy to it.