I've heard that $Sp$ is analogous to the derived category $Der(Ch\mathbb{Z})$ (I will thus refer to those two categories as the left and right side respectively below).
Namely, every spectra is a colimit of shifts of $\mathbb{S}$ which is thus analogous to $\mathbb{S}$ being $\mathbb{Z}$ concentrated at $0$.
Likewise smash product with $\mathbb{S}$ is the identity on the left which confirms the analogy yet again.
However, $KG(R)$ modules on the left (under the smash product) should be analogous in some way to $R$ chain complexes on the right. In fact, the Postinokov tower shows we can literally write every spectrum (analogous to chain complex) as the limit of trimmed chain complexes with fibers $KG(H)$ for abelian groups $H$. This is intuitively analogous to writing any complex on the right as a limit of zeroing out less and less of the terms.
In conclusion I'm confused about the comparison because in $Sp$ there are both $KG(R)$ and $\mathbb{Z}$ and would like clarification
The approach in Axiomatic Stable Homotopy Theory (Hovey, P, Strickland) suggests several analogies: one is that the homotopy category of spectra is analogous to the derived category of a commutative ring like $\mathbb{Z}$, but in this setting, homotopy and homology coincide. I think this is the issue you're raising. So maybe the derived category of $\mathbb{Z}$ is analogous to a version of spectra in which the sphere and $H\mathbb{Z}$ coincide.
Another analogy is between the homotopy category of spectra and an appropriate category of chain complexes of modules over a cocommutative Hopf algebra, for example the group ring $kG$ of a $p$-group $G$. In this case you want to consider the category of chain complexes of projective modules, up to chain homotopy. In this case homotopy and homology are different, and there are interesting chain complexes with zero homology, so the derived category doesn't seem to be the right analogy.