I would never say, and don’t recall ever hearing, “potato-potahto”. Moreover, I don’t think it’s a good way to describe two things that are the same.
The referenced song, a love song, is saying that people with different perspectives can still fall in love. (The fact that it uses nonsense examples (no one says “potahto”) is typical of the songwriter’s style - catchy, whimsical, but often nonsensical.) So in the original song, it’s not expressing equality. It’s talking about overlooking differences in background or expression.
To express general equality, more common is “six of one, half dozen of another”