**Edit.** User hic points out that with the first action I specified, there are infinitely many orbits. So please change the group to $\textbf{PGL}_2\times \textbf{SL}_2$ with its left-right action.

The examples in my comment are only quasi-projective, not projective. However, there are also projective examples. Begin with the $4$-dimensional vector space $\text{Mat}_{2\times 2}$ of $2\times 2$-matrices, with its usual matrix product. Let $\mathbb{P}\text{Mat}_{2\times 2}$ denote the associated projective $3$-space. Denote by $\textbf{PGL}_2\subset \mathbb{P}\text{Mat}_{2\times 2}$ denote the open subset of invertible $2\times 2$ matrices considered up to scaling. **Edit.** *Also let $\textbf{SL}_2\subset \text{Mat}_{2\times 2}$ denote the degree $2$ cover of $\textbf{PGL}_2$; the group of $2\times 2$ matrices with determinant $1$. Let $G$ be $\textbf{PGL}_2\times \text{SL}_2$.* For every action of $G$ on a scheme $Z$, since the character group of $G$ is trivial, the natural homomorphism $$\text{Pic}^{G}(Z) \to \text{Pic}(Z)$$ is injective. This raises the question: what is the image?

The left action of $\textbf{PGL}_2$ on itself extends to a left action of $\textbf{PGL}_2$ on $\mathbb{P}\text{Mat}_{2\times 2}$. **Edit.***The right action of $\textbf{SL}_2$ on $\textbf{PGL}_2$ extends to a right action of $\textbf{SL}_2$ on $\mathbb{P}\text{Mat}_{2\times 2}$. Together this defines an action of $G$ on $\mathbb{P}\text{Mat}_{2\times 2}$, $([g],h)\cdot [A] = [g\cdot A\cdot h^{-1}]$. This action has two orbits, the open orbit $\textbf{PGL}_2$ and the closed orbit is the zero locus of the determinant. In particular, the closed orbit is a $G$-invariant Cartier divisor whose associated invertible sheaf is isomorphic to $\mathcal{O}(2)$ with a natural $G$-linearization.* The image of the natural group homomorphism $$\text{Pic}^{G}(\mathbb{P}\text{Mat}_{2\times 2}) \to \text{Pic}(\mathbb{P}\text{Mat}_{2\times 2}) $$ is the index $2$ subgroup generated by the class $[\mathcal{O}(2)]$. Now set $X=Y=\mathbb{P}\text{Mat}_{2\times 2}$. The Künneth homomorphism $$\text{Pic}(\mathbb{P}\text{Mat}_{2\times 2}) \times \text{Pic}(\mathbb{P}\text{Mat}_{2\times 2}) \to \text{Pic}(\mathbb{P}\text{Mat}_{2\times 2} \times \mathbb{P}\text{Mat}_{2\times 2} ) $$ is an isomorphism. Thus the image of the injective group homomorphism $$\text{Pic}^{G}(\mathbb{P}\text{Mat}_{2\times 2}) \times \text{Pic}^{G}(\mathbb{P}\text{Mat}_{2\times 2}) \to \text{Pic}(\mathbb{P}\text{Mat}_{2\times 2} \times \mathbb{P}\text{Mat}_{2\times 2} ) $$ is the subgroup generated by $([\mathcal{O}(2)],0)$ and $(0,[\mathcal{O}(2)])$.

Now for elements $([A_1],[A_2]) \in \mathbb{P}\text{Mat}_{2\times 2}\times \mathbb{P}\text{Mat}_{2\times 2}$ with coordinates $$A_1 = \left[ \begin{array}{cc} a_1 & b_1 \\ c_1 & d_1 \end{array} \right], \ A_2 = \left[ \begin{array}{cc} a_2 & b_2 \\ c_2 & d_2 \end{array} \right],$$ consider the matrix product of the adjugate of $A_2$ and $A_1$, $$ B = \text{adj}(A_2)\cdot A_1 = \left[ \begin{array}{cc} d_2 & -b_2 \\ -c_2 & a_2 \end{array} \right] \cdot \left[ \begin{array}{cc} a_1 & b_1 \\ c_1 & d_1 \end{array} \right] = \left[ \begin{array}{cc} a_1d_2-b_2c_1 & b_1d_2-b_2d_1 \\ a_2c_1-a_1c_2 & a_2d_1-b_1c_2 \end{array} \right].$$ Then entries of the matrix $B$ generate a subspace $U$ of the space $$ V=H^0(\mathbb{P}\text{Mat}_{2\times 2} \times \mathbb{P}\text{Mat}_{2\times 2}, \text{pr}_1^*\mathcal{O}(1)\otimes \text{pr}_2^* \mathcal{O}(1)).$$ The left, resp. right, diagonal action of $\textbf{SL}_2$ on $\mathbb{P}\text{Mat}_{2\times 2} \times \mathbb{P}\text{Mat}_{2\times 2}$ lifts to a unique linearization of $\text{pr}_1^*\mathcal{O}(1)\otimes \text{pr}_2^* \mathcal{O}(1)$. The subspace $U$ is a left-right $\textbf{SL}_2\times \textbf{SL}_2$-subrepresentation of $V$ that is **left trivial**, i.e., for every $g\in \textbf{SL}_2$, $$\text{adj}(g\cdot A_2)\cdot (g\cdot A_1) = \left( \text{adj}(A_2)\text{adj}(g)\right)\cdot (g\cdot A_1) = $$ $$\text{adj}(A_2)\left( \text{adj}(g)g \right) A_1 = \text{adj}(A_2)\left( \text{Id}_{2\times 2}\right) A_1= \text{adj}(A_2)A_1.$$ **Edit.***Thus, the left-right action of $\textbf{SL}_2\times \textbf{SL}_2$ factors through an action of $\textbf{PGL}_2\times \textbf{SL}_2$. ***Edit.**In particular, the nonzero element in $U$ corresponding to the trace of $B$ is invariant under $\textbf{PGL}_2\times \textbf{SL}_2$. The corresponding zero divisor of this section of $\text{pr}_1^*\mathcal{O}(1)\times \text{pr}_2^*\mathcal{O}(1)$ is a $G$-invariant divisor. Thus, the invertible sheaf of this divisor has a natural $G$-linearization. Therefore $([\mathcal{O}(1)],[\mathcal{O}(1)])$ is an element in the image of $\text{Pic}^{G}(\mathbb{P}\text{Mat}_{2\times 2}\times \mathbb{P}\text{Mat}_{2\times 2})$ that is not in the image of $\text{Pic}^{G}(\mathbb{P}\text{Mat}_{2\times 2}) \times \text{Pic}^{G}(\mathbb{P}\text{Mat}_{2\times 2}).$ Thus the Künneth homomorphism $$\text{Pic}^{G}(\mathbb{P}\text{Mat}_{2\times 2}) \times \text{Pic}^{G}(\mathbb{P}\text{Mat}_{2\times 2}) \to \text{Pic}^{G}(\mathbb{P}\text{Mat}_{2\times 2} \times \mathbb{P}\text{Mat}_{2\times 2} ) $$ is not surjective.