For two matroids $M_1$ and $M_2$ with the same ground set $V$ and two cost functions $w_1$ and $w_2$ on $2^V$, we consider the problem of finding bases $X_1$ of $M_1$ and $X_2$ of $M_2$ minimizing $w_1(X_1)+w_2(X_2)$ subject to a certain cardinality constraint on their intersection $X_1 \cap X_2$. For this problem, Lendl, Peis, and Timmermans (2019) discussed modular cost functions: they reduced the problem to weighted matroid intersection for the case where the cardinality constraint is $|X_1 \cap X_2|\le k$ or $|X_1 \cap X_2|\ge k$; and designed a new primal-dual algorithm for the case where the constraint is $|X_1 \cap X_2|=k$. The aim of this paper is to generalize the problems to have nonlinear convex cost functions, and to comprehend them from the viewpoint of discrete convex analysis. We prove that each generalized problem can be solved via valuated independent assignment, valuated matroid intersection, or $\mathrm{M}$-convex submodular flow, to offer a comprehensive understanding of weighted matroid intersection with intersection constraints. We also show the NP-hardness of some variants of these problems, which clarifies the coverage of discrete convex analysis for those problems. Finally, we present applications of our generalized problems in the recoverable robust matroid basis problem, combinatorial optimization problems with interaction costs, and matroid congestion games.