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This phenomenon is known as diplacusis and there are several different flavors.
Diplacusis binauralis occurs when you hear the same sound differently in each ear. This condition is also known as interaural pitch difference IPD. The why and how is complex and more info can be obtained here. Related to diplacusis binauralis, diplacusis dysharmonica is when the sound is perceived normally in one ear and at a different pitch to the other.
Diplacusis echoica is when you hear the same sound repeated as an echo in the affected ear. This condition is typically due to middle ear pathology.
Diplacusis monauralis occurs when one ear hears the same sound as two different sounds at the same time.
Treatment depends on what type of diplacusis is present and localizing the source of the problem leading to the diplacusis. Typically, this would entail an examination of the ear followed by a hearing test. At that point, treatment or further testing may be required.
More info on this condition can be found here.