Sound event localization and detection (SELD) involves identifying the direction-of-arrival (DOA) and the event class. The SELD methods with a class-wise output format make the model predict activities of all sound event classes and corresponding locations. The class-wise methods can output activity-coupled Cartesian DOA (ACCDOA) vectors, which enable us to solve a SELD task with a single target using a single network. However, there is still a challenge in detecting the same event class from multiple locations. To overcome this problem while maintaining the advantages of the class-wise format, we extended ACCDOA to a multi one and proposed auxiliary duplicating permutation invariant training (ADPIT). The multi- ACCDOA format (a class- and track-wise output format) enables the model to solve the cases with overlaps from the same class. The class-wise ADPIT scheme enables each track of the multi-ACCDOA format to learn with the same target as the single-ACCDOA format. In evaluations with the DCASE 2021 Task 3 dataset, the model trained with the multi-ACCDOA format and with the class-wise ADPIT detects overlapping events from the same class while maintaining its performance in the other cases. Also, the proposed method performed comparably to state-of-the-art SELD methods with fewer parameters.