It seems to me that your question is actually about something that is clearer in a more basic context than chain complexes. Fix a field $k$ and let's work instead in the category $C$ of pairs $(V, T)$ where $V$ is a vector space and $T : V \to V$ is a linear transformation. This category, as a $k$-linear category, admits the following two descriptions:

• It is the category of $k$-linear representations of the monoid $\mathbb{N}$.
• It is the category of $k$-linear representations of a one-dimensional Lie algebra over $k$.

These two descriptions lead to two different notions of tensor product (that is, two monoidal structures): for one, the tensor product $V_1 \otimes V_2$ is acted on by $T_1 \otimes T_2$, but for the other, the tensor product $V_1 \otimes V_2$ is acted on by $$T_1 \otimes I + I \otimes T_2.$$

(This is necessary in order for the action to exponentiate to the correct action of the corresponding Lie group when $k = \mathbb{R}$, for example.) Abstractly, both descriptions above say that $C$ is equivalent to the category of modules over $k[x]$, the first description via the construction of the monoid algebra and the second description via the construction of the universal enveloping algebra. However, the natural notions of tensor product in each setting is different: both the monoid algebra and universal enveloping algebra constructions actually spit out bialgebras, and the comultiplication on each bialgebra gives rise to the tensor product. The first construction spits out $k[x]$ with the comultiplication $$x \mapsto x \otimes x$$

whereas the second construction spits out $k[x]$ with the comultiplication $$x \mapsto x \otimes 1 + 1 \otimes x.$$

So what you've observed is that the category of chain complexes is morally closer to the second construction than the first. Why this is the case is something that I don't have a completely satisfactory answer for, but in any case the point I want to make is that the second comultiplication is just as natural as the first from the appropriate perspective.

Algebro-geometrically, comultiplications on $k[x]$ correspond to algebraic monoidal structures on the affine line $\mathbb{A}^1$. The first construction comes from multiplication and the second construction comes from addition.

It seems to me that your question is actually about something that is clearer in a more basic context than chain complexes. Fix a field $k$ and let's work instead in the category $C$ of pairs $(V, T)$ where $V$ is a vector space and $T : V \to V$ is a linear transformation. This category, as a $k$-linear category, admits the following two descriptions:

• It is the category of $k$-linear representations of the monoid $\mathbb{Z}_{\ge 0}$.
• It is the category of $k$-linear representations of the one-dimensional Lie algebra over $k$.

These two descriptions lead to two different notions of tensor product (that is, two monoidal structures): for one, the tensor product $V_1 \otimes V_2$ is acted on by $T_1 \otimes T_2$, but for the other, the tensor product $V_1 \otimes V_2$ is acted on by $$T_1 \otimes I + I \otimes T_2.$$

(This is necessary in order for the action to exponentiate to the correct action of the corresponding Lie group when $k = \mathbb{R}$, for example.) Abstractly, both descriptions above say that $C$ is equivalent to the category of modules over $k[x]$, the first description via the construction of the monoid algebra and the second description via the construction of the universal enveloping algebra. However, the natural notions of tensor product in each setting is different: both the monoid algebra and universal enveloping algebra constructions actually spit out bialgebras, and the comultiplication on each bialgebra gives rise to the tensor product. The first construction spits out $k[x]$ with the comultiplication $$x \mapsto x \otimes x$$

whereas the second construction spits out $k[x]$ with the comultiplication $$x \mapsto x \otimes 1 + 1 \otimes x.$$

So what you've observed is that the category of chain complexes is morally closer to the second construction than the first. Why this is the case is something that I don't have a completely satisfactory answer for, but in any case the point I want to make is that the second comultiplication is just as natural as the first from the appropriate perspective.

Algebro-geometrically, comultiplications on $k[x]$ correspond to algebraic monoidal structures on the affine line $\mathbb{A}^1$. The first construction comes from multiplication and the second construction comes from addition.