In constraint systems for ZK proofs, why are multiplications counted but are additions not?
Your question seems to assume this is true for any constraint proof system. I'm quite confident that this property is on a case-by-case basis. Bulletproofs, for example, have this property: proof size is $\log(n)$, with $n$ the amount of multipliers in the arithmetic circuit. In the rest of my answer, I'll talk about how a bulletproof achieves this.

I'll give a very condensed sketch of how bulletproofs achieve this property. For a more in-depth explanation, I suggest to read the above paper, or the (in my opinion) simpler notes of the good folks at dalek-cryptography.

Bulletproofs writes their constraint system in a single vector equation like follows:

\begin{aligned}
\mathbf{W}_L \cdot \mathbf{a}_L +
\mathbf{W}_R \cdot \mathbf{a}_R +
\mathbf{W}_O \cdot \mathbf{a}_O =
\mathbf{W}_V \cdot \mathbf{v} +
\mathbf{c}
\end{aligned}

Here, the $\mathbf{W}_{\{L,R,O,V\}}$ are weight matrices: they map secrets from their right hand side to constraints. $\mathbf{v}$ are the input secrets. Note that this equation contains the linear constraints, not the multipliers. In what follows, they combine it with the multipliers ($\mathbf{a}_L\circ \mathbf{a}_R=\mathbf{a}_O$), and then this equation is molded into a form $t(x)=\langle\mathbf{l}(x),\mathbf{r}(x)\rangle$, in which $x$ is a challenge of the verifier. Here's the first clue: $\mathbf{l}(x)$ and $\mathbf{r}(x)$ are of length $n$, the amount of multiplication gates.

The proof process links $t(x)$ and the two vectors $\mathbf{l}(x)$ and $\mathbf{r}(x)$ to the secret input commitments (in constant size), and then goes on to prove that their inner product relation holds: the inner product proof. This was introduced by Bootle et al., in Efficient Zero-Knowledge Arguments for Arithmetic Circuits in the Discrete Log Setting, and improved upon in the bulletproofs paper. This is your second clue: this inner product argument that proves something of the form $t=\langle \mathbf{l}, \mathbf{r}\rangle$ needs $\log(|l|)=\log(|r|)$ communication.