It's called Middle C because it's the C in the middle of the keyboard.
All the other Cs are called C because we perceive them as "the same note."
Ask her to play a tune, and then play it again an octave higher and then lower. The pitch is different, but there's also a sense that all three versions are the same.
Mathematically, pitch is like a helix. Points with the same rotation - which happen to be a frequency ratio of powers of 2 - sound very closely related.
Instead of giving every single note a different name, it's useful to give notes the same repeating set of 12 names and then add a separate "height" variable - called the octave - to describe which turn of the helix they're on.
If you play pitches with the same rotation at the same time they blend well together. In fact you can barely tell them apart.
Of course you can rotate by less than a full circle. If you play the same tune with a different pitch offset - say one note instead of twelve, which is the smallest rotation you can do on a conventional keyboard - you'll hear it's still the same shape, but there's a jarring effect between the different versions.
Some rotations sound good together, but not as good as the identity rotation. Others clash and sound bad.
As a guide, simple divisions of the pitch circle - like 3:2 - sound smoother than the more complicated divisions.
The twelve notes approximate the rotations that sound best together. You get a good mix of smooth and not so smooth blends, and you can build all kinds of combinations with them.
You use approximations and not exact whole number ratios because you want to be able to write music that isn't stuck with a single starting/reference note. You can play the same tune at different starting pitches and the relationships between the notes doesn't change.
If you use whole number ratios a tune that sounds good starting on one note sounds weird and sour starting on others. The pitch spiral turns into an oval corkscrew, and the neat rotations get distorted.
And yes, all the As, Bs, Cs have their own specific frequencies. By convention A = 440Hz, and everything else is tuned around that. All the As are powers of 2 either side of 440Hz - from 27.5Hz, which is the lowest note on a piano, to 3520Hz for the highest A.