While the basic greedy algorithm gives a semi-streaming algorithm with an approximation guarantee of $2$ for the \emph{unweighted} matching problem, it was only recently that Paz and Schwartzman obtained an analogous result for weighted instances. Their approach is based on the versatile local ratio technique and also applies to generalizations such as weighted hypergraph matchings. However, the framework for the analysis fails for the related problem of weighted matroid intersection and as a result the approximation guarantee for weighted instances did not match the factor $2$ achieved by the greedy algorithm for unweighted instances. Our main result closes this gap by developing a semi-streaming algorithm with an approximation guarantee of $2+ε$ for \emph{weighted} matroid intersection, improving upon the previous best guarantee of $4+ε$. Our techniques also allow us to generalize recent results by Levin and Wajc on submodular maximization subject to matching constraints to that of matroid-intersection constraints. While our algorithm is an adaptation of the local ratio technique used in previous works, the analysis deviates significantly and relies on structural properties of matroid intersection, called kernels. Finally, we also conjecture that our algorithm gives a $(k+ε)$ approximation for the intersection of $k$ matroids but prove that new tools are needed in the analysis as the used structural properties fail for $k\geq 3$.