You could try to use multiple models, where each model predicts a single class vs. others, but would then need to decide how to use the results. I.e. if you are working on a multi-label classification (zero, one, or multiple classes can be active for each sample), you could directly use the output. On the other hand, if you are working on a multi-class classification, you would have to decide, if e.g. only the highest prediction should be used as the class prediction.
Usually you would just use a single model and let it predict multiple classes directly.