We discuss the isospin symmetry breaking effects on threshold cusp structures in multichannel scattering near two-body thresholds. In hadronic systems with isospin symmetry, two or more nearly degenerate thresholds can appear, and their small splitting due to isospin breaking can generate multiple cusp structures in a narrow energy region. In this paper, using the $K$-matrix representation, we derive a general expression for the scattering amplitude near the thresholds and show that the cusp structures can be classified by the signs of the slopes of the cross section above and below threshold. We also show that additional restrictions appear in two- or three-channel systems and in the Flatt'e amplitude. For three-channel scattering with two nearby thresholds, we clarify how the two cusp structures are related when the threshold splitting is small and how they merge into a single cusp in the degenerate limit. Finally, we discuss the cusp structures in the $\Lambda p$ elastic cross section in the coupled $\Lambda p$-${\Sigma }^{+}n$-${\Sigma }^{0}p$ system. We show that, when isospin breaking is small, the properties of the two cusp structures are constrained by isospin symmetry. We also perform quantitative calculations using both simplified examples and realistic input based on N${}^2$LO chiral effective field theory, and find that isospin breaking can significantly modify the relative sharpness of the cusps and may even change the cusp type itself.