Let $G$ and $H$ be two primitive unitary groups, is $G\times H$ necessarily a primitive unitary group? If not, is there any counterexample?
I try to give an answer based on my interpretation of your question. I guess by "unitary" group you mean "subgroup of a unitary group", because $U(n)\times U(m)$ is certainly not isomorphic to $U(k)$ for any $k$. Then any finite group is fine for an example. Whenever a group $G$ acts primitively on a set $X$, it acts transitively and therefore the cardinalty of $X$ is less than the order of $G$. When $G$ acts primitively on $X$, the cardinality of $X$ is called the degree of $G$. The only finite primitive group of degree $2$ is $S_2$. Assume $S_2\times S_2$ is primitive. Then the degree of $S_2\times S_2$ must be less than $4$. The only primitive finite groups of degree $3$ are $S_3$ and $A_3$. The only ones of degree $4$ are $S_4$ and $A_4$, according to the List of small primitive groups implemented in GAP. All are not isomorphic to $S_2\times S_2$, so the answer to your question is no (assuming my interpretation is correct).