Check https://en.wikipedia.org/wiki/Pythagorean_comma.

Historically, they sounded different, so they wouldn't even be acoustically in-tune w/ the same notes. Each would only be in tune with its nearby notes on the "extended" cycle of fifths.

What does this mean for a piano keyboard, for example? Is the fact that it's the same key related to the fact that a piano cannot be perfectly tuned?

Yes, and it means that songs actually sound different when played in different keys. It's not just a shift up or down, the intervals within the song shift, if only slightly.

They're slightly different pitches in a non-equal temperament, depending on the key the temperament is based on.

Indeed "slightly different", as opposed to "totally different", which they are not.

And apparently the human ear can be train to ignore this slight difference (which everybody does because we are used to 12-TET). And this, for me, kind of throws the whole "simple integer fraction ratio == pleasing harmony" a bit into question. It's probably not wrong, but there's definitely more to it. But it's hard to explore, because you need the exposure to get used to the new microtonals if you want to experiment with it. Definitely very hard to test scientifically because it depends so much on a particular person's musical background and education.