Cohomological dimension of $G \times G$

One fairly general class of groups $G$ for which $\operatorname{cd}(G\times G)=2\operatorname{cd} (G)$ are the duality groups. A group $G$ is a duality group of dimension $n$ if there is a dualizing $G$-module $D$ and a fundamental class $e\in H_n(G;D)$ such that $$ -\cap e :H^k(G;M)\to H_{n-k}(G;M\otimes D) $$ is an isomorphism for any $k$ and any $G$-module $M$. It follows that $\operatorname{cd}(G)=n$. If the underlying group of $D$ can be taken as $\mathbb{Z}$, then $G$ is a Poincaré duality group. There are duality groups which are not Poincaré duality groups, and therefore can't have a closed aspherical manifold as their classifying space. Finitely generated free groups and knot groups are examples.

A theorem of Bieri and Eckmann states that given an extension $$ 1\to K\to G\to Q\to 1 $$ iwhere $K$ and $Q$ are duality groups of dimensions $m$ and $n$ respectively, then $G$ is a duality group of dimension $m+n$. This implies the claim at the top of my answer. This result and many more useful facts about duality groups can be found in the reference

Bieri, Robert; Eckmann, Beno, Groups with homological duality generalizing Poincaré duality, Invent. Math. 20, 103-124 (1973). ZBL0274.20066.

If $G$ is a non-trivial group with $G\times G\cong G$, then $G$ has infinite cohomological dimension as @YCor suggests. If $H\leq G$, then $cd(H)\leq cd(G)$ because $\mathbb ZG$ is a free $\mathbb ZH$-module and so any $\mathbb ZG$-free resolution of the trivial module is a $\mathbb ZH$-free resolution. If $G$ has an element of finite order, then $G$ has infinite cohomological dimension because finite cyclic groups do. If $G$ has an element of infinite order, then $G\cong G\times G$ implies $\mathbb Z^n$ is a subgroup of $G$ for all $n>0$. Since $cd(\mathbb Z^n)=n$, this means $G$ that $cd(G)\geq n$ for all $n>0$ and hence is infinite.

Another class of groups for which $cd(G\times G)=2 cd(G)$, containing duality groups, is the class of groups of type $FP_{\infty}$.

If $G$ is of type $FP_{\infty}$ that is the trivial module has a projective resolution that is finitely generated in each degree, then a theorem of Ken Brown https://www.math.cornell.edu/~kbrown/scan/1975.0050.pdf implies that the functor $H^n(G,-)$ commutes with direct limits for all $n$. Thus if $G$ has cohomological dimension $k$ then there is a finitely generated $G$-module $A$ with $H^k(G,A)\neq 0$ (there is some module for which this cohomology doesn't vanish and any module is a direct limit of its finitely generated submodules). So your Kunneth argument should give $cd(G\times G)=2 cd(G)$ if $G$ is of type $FP_{\infty}$.

This includes the example of duality groups by Corollary 1 of Brown's paper although his argument is essentially Mark Grant's.