While fishes may not have obvious personalities like any other pets, each fish species could display a wide spectrum of traits. Their social interaction and complex personalities become evident when housed together in groups with the same or compatible breeds.
There are, however, some breeds that are semi-aggressive hence they should not be tanked with other fish. In contrast, goldfish are usually affable when housed together with their own species.

It should be noted that different fish within the same species may react to stress and stimuli differently. Different breeds also require varying water parameters (e.g. pH, KH/GH, temperature) to thrive. Therefore, you should get to know the temperaments of each fish to decide their compatibility with other fishes, and suitability for your lifestyle and habits.