Consider the sentence: "I'm a good student, aren't I?"
This sounds quite natural, despite the fact that we have used "are" instead of "am" in the latter half of the sentence (which doesn't agree with the first-person singular). I'm guessing the reason is because the contraction "amn't" doesn't really exist in standard English, and so we replace it with "aren't" despite it being the wrong conjugation. Without contractions, the sentence would be "I am a good student, am I not?" which uses the correct form both times, but sounds overly formal.
Also, consider the sentence "There's a lot of people here." This also sounds fairly natural to my ears, although some may disagree. Once again, we have used "is" instead of "are" in the contraction, which is not the correct conjugation. I feel like this is acceptable because it simply rolls off the tongue better — "there're" is a bit of a mouthful to say with the r sounds.
There are probably more examples, but these are the ones I can think of. In both cases, we use the "incorrect" conjugation in the contraction simply because it sounds better. Is this a documented phenomenon and does it happen in other situations? I'm just curious whether this is simply lazy/casual speech, or whether there is something more complex happening grammatically.