I am considering the space $A$ consisting of the Kronecker product: \begin{eqnarray} A=\{U(N)\otimes U(k)\}, \end{eqnarray} where $U(N)$ is the $N$ by $N$ unitary matrices and $\otimes$ denotes the Kronecker product of matrices.
My question is whether we can find a homeomorphism between $A$ and some direct product of Lie group in the form as \begin{eqnarray} U(1)\times\cdots\times U(1)\times SU(N)\times SU(k). \end{eqnarray} Here, to distinguish from Kronecker product $\otimes$, we use $\times$ to denote the direct product of groups.
I believe the Kronecker product is the same thing as the tensor product, represented relative to a basis?
But in general tensor product is something we do to linear spaces, not Lie groups. So I don't really have an answer to your question, except to say that it doesn't seem like a very natural question. Consider this perhaps an extended comment instead of an attempted answer.
However remember that $U(n)$ is defined as a set of matrices which preserve some hermitian form on a vector space. It definitely makes sense to take tensor products of those vector spaces, and see how the matrices act on the resulting space. The decomposition is a little involved, going under the name of Clebsch-Gordon coefficients.