First of all, Alama - Formal proofs and refutations contains information about Euler's Polyhedron Formula. If you look at pp. 47–49, you will see that Poincaré proved Euler's Polyhedron Formula, and the following information appears there.

The property of being a homology sphere is the property that $B_k = Z_k$, that the $k$-circuits are the bounding $k$-chains. The inclusion $B_k \subseteq Z_k$ says that $\partial_{k+1}\partial_k \equiv 0$.

From this, we can see that Poincaré's proof of Euler's Polyhedron Formula has some relation to the homology sphere.

And also if you read Lamb - A few of my favorite proofs: The Poincaré homology sphere,

Early topologists wanted to try to find ways of distinguishing spaces by finding invariants: numbers or other mathematical objects that could be assigned to each space. Ideally, two spaces with the same invariant would be the same space, and two spaces with different invariants would be different spaces. Poincaré came up with Betti numbers, which are informally a way to catalogue the holes of different dimensions in a space, and torsion coefficients, which sort of keep track of twistedness. In a 1900 paper, Poincaré conjectured that these Betti numbers and torsion coefficients (also known today as homology) could tell you whether or not a space was a sphere.

When Poincaré thought about homology, we can see that he was also thinking about spheres.

Of course, the book Euler's Gem p.254 also explains that homology originated from Riemann's connectivity and Betti's higher-dimensional generalizations, but while reading the papers and articles mentioned above, I wondered if Euler's Polyhedron Formula might have also contributed to the emergence of homology.