The effects of the Coulomb interaction on the energy spectrum and the magnetization of two electrons in a strained Inx Ga1-x As/GaAs ringlike nanostructure are analyzed with realistic parameters inferred from the cross-sectional scanning-tunneling microscopy data. With an increasing magnetic field, the lowest spin-singlet and spin-triplet states sequentially replace each other as the ground state. This is reminiscent of the Aharonov-Bohm effect for the ringlike structures. The exchange interaction leads to a more complicated oscillatory structure of the magnetic moment of the two electrons as a function of the magnetic field as compared to the magnetization pattern for a single-electron ringlike nanostructure. We discuss the relevance of the two-electron systems for the interpretation of the Aharonov-Bohm oscillations in the persistent current observed in low temperature magnetization measurements on self-assembled Inx Ga1-x As/GaAs ringlike nanostructures.