I think most statisticians would agree that overfitting isn't necessarily due to just high variance, but rather typically from having a model that is too flexible and not having enough data to constrain the flexibility after seeing the data.
While it is often described as the model being so flexible it just fits the noise, hence I believe why you may think overfitting is just a function of high variance, I can provide a trivial example with no noise that still can suffer from overfitting.
Suppose you have a deterministic linear relation $y = x + 1$. If you have only two points, and you foolishly decide to fit a quadratic equation to those two points, any quadratic function that passes between those two observed points will perfectly fit the data, yet will almost surely make incorrect inference at any two points besides the two observed. Note that the conditional variance in the data in this case is 0!