I have a matrix $M$ of the form $M_{(ij),(pq)} = A_{i,p} B_{j,q}$, where $A,B$ are two square matrices. I need to compute the eigenvalues and eigenvectors of $M$.
If $\alpha, u$ and $\beta, v$ are an eigenvalue/eigenvector pair for $A$ and $B$, respectively, then it is easy to see that the vector $w_{(ij)} = u_i v_j$ is an eigenvector of $M$ with eigenvalue $\alpha\beta$.
Are all eigenvalue/eigenvectors of $M$ of this form?
No because if you have pairs $v_1,v_2$ and $u_1,u_2$ of independent eigenvectors with the same eigenvalues or at least $\alpha_1\beta_1 = \alpha_2\beta_2$ then $$u_1\otimes v_1 + u_2\otimes v_2$$ is an $\alpha\beta$ eigenvector as well but is not a pure tensor.
On the otherhand it is in the span of eigenvectors of the form which you mentioned, and so long as your field is algebraically closed so that there are bases of eigenvectors for $A$ and $B$, then the product of the bases gives a basis of eigenvectors for $M$.