Why does mirror flip the left and right, but not the top and bottom ?

The answer is it does not !

What mirror does is to change the longitudinal coordinate, but not the lateral ones. Let's say the mirror is place in x-y place, at origin z=0. Then any point at coordinate (x,y,z) will be mapped to (x,y,-z). Only the longitudinal coordinate z is changed. If you stand facing a mirror and place an object between you and the mirror. The closer to you the object is, the further away is the image. The object to the left is still to the left, same goes with right, top and bottom.

Now, if you wear a T-shirt, with a word on it, let's say, it is "TOYOTA", in the mirror, you will read "ATOYOT" on your shirt. And that is where the "rumor" of "flipping" comes :-) In fact, the mirror did not change the left to right. When you wear the T-shirt, from your perspective, the letters are indeed ATOYOT, reading from left to right; only that you can not see it easily without a mirror. One can verify it by pulling the T-shirt up by the collar, when the letters are at the eye level and the T-shirt fabric is thin enough, you will see it. What the mirror did is to move the letters longitudinally, so you can read it easily.