My only reference for this topic has been the following nlab article https://ncatlab.org/nlab/show/interval+object+in+chain+complexes
This should all work for any closed monoidal abelian category. But in particular for $\mathrm{Ch}(R-\text{Mod})$ the notion of a tensor product is defined as.
For $A, B \in \mathrm{Ch}(R-\text{Mod})$ define $A \otimes B$ component wise as
$$ (A\otimes B)_n=\oplus_{n=i+j} A_i \otimes B_j $$
with the differential as $$ \varphi_n(a,b)=(\varphi_Aa\otimes b)+(-1)^{\mathrm{deg}(a)}(a \otimes \varphi_B b)$$
n-Lab defines a interval object in $\mathrm{Ch}(R-\text{Mod})$ as the following complex, $$ 0 \to 0 \to R \to R\oplus R $$
This allows for nicely constructing object cylinder as $I \otimes A$ and the concept of mapping cones and mapping cylinders follows as pullback consctructions from this. This also gives us the Puppe sequence for chain complexes with `suspension' being shifting the complex by a minus 1 degree and adding a minus sign to the differential.
What motivates this definition of the interval object in the category of chain complexes? nLab unfortunately didn't have any references with which I could read more about it. It mentions it as a special case of a "normalized chain complex of the simplicial chains on the simplicial 1-simplex:" but this reads like a word salad to me. I don't understand clearly what its connection is at all.
I don't know much about homotopy theory but as far as model categories go it doesn't use a notion of an interval.
Question 1: What motivates the particular definition of interval object given in the nLab article?
Question 2: Can we work formally in the language of homotopy theory? i.e. are interval objects needed to derive, say the homotopy cofiber sequence categorically? Or can we do it without them
This may not be what you expect but I still put it as an answer. From my experience, if you want to do homotopy theory (or at least model categories), it is good to have notions of cylinder objects and path objects. The best situation is to have an interval and then as you noticed, we can tensor the interval to creat cylinder objects.
Now if you know some model categories, then you may know that the homotopy theory of chain complexes and the homotopy theory of simplicial sets are not so different. The reason lies in the famous theorem called Dold-Kan correspondence. There is an equivalences of categories, with one direction is the normalized complex $$N \colon \mathbf{Ab}^{\Delta^{op}} \longrightarrow \mathbf{Ch}_{\geq 0}(\mathbb{Z})$$ where the LHS is the category of simplicial abelian groups, and the RHS is the category of complexes of $\mathbb{Z}$-modules (you can replace $\mathbb{Z}$ with any ring, or even abelian categories). There is also an adjunction $$(\text{free},\text{forget}) \colon \mathbf{Set}^{\Delta^{op}} \longrightarrow \mathbf{Ab}^{\Delta^{op}}$$ where LHS is simplicial sets, given by the free functor and the forgetful functor between sets and abelian groups. If you can convince yourself that simplicial sets are not different from topological spaces (in fact, their homotopy categories are equivalence), then the unit interval in $\mathbf{Set}^{\Delta^{op}}$ "should" be $\Delta[1]$ and therefore the interval in $\mathbf{Ab}^{\Delta^{op}}$ "should" be $\mathbb{Z}[\Delta[1]]$. In terms of the Dold-Kan correspondence, the unit interval of $\mathbf{Ch}(\mathbb{Z})$ "should" be $N(\mathbb{Z}[\Delta[1]])$.