In Pugh's Real Mathematical Analysis he writes:
The quotient vector space $$H^k(U) = Z^k(U)/B^k(U)$$ is called the $k$th de Rham cohomology group of $U$. Its members are the "cohomology classes" of $U$.
While of course there's a forgetful functor $\operatorname{Vect}_\mathbb{R}\to\operatorname{Grp}$, and we don't really need the scalars to do homology, it seems like a bit of a fraud to call something a group that we've constructed as a vector space, and whose most interesting property is its dimension.
In these notes on homology a $k$-chain is defined as a member of the free abelian group $Q_k$ of finite linear combinations of $k$-cubes. But it then talks about subspaces, rank, and bases as if $Q_k$ really was a vector space after all.
Why do we call these vector spaces groups?
It's just traditional. The first kinds of (co)homology objects to be studied were abelian groups (specifically, simplicial homology groups), and the name stuck in some other contexts even when there is more structure than just a group.