Why are plane mirror images laterally inverted and not vertically inverted?

It actually has little to do with the plane mirror.

Suppose I say, "Show me that a mirror image is laterally inverted."

What you would probably do is:

Step 1: Show me two copies of a piece of paper with the letter 'b'
Step 2: Place one of them in front of a plane mirror, with the letter 'b' facing the plane mirror
Step 3: Place the other one next to the mirror image, with the letter 'b' facing us.

Then, it is easy to see, that the mirror image shows the letter 'd' while the paper shows the letter 'b'.

However, let us examine what you did in step 3: you rotated the piece of paper about the vertical axis. Is that the only way which you can make the piece of paper face yourself so that you can read it? Clearly not. You could rotate the piece of paper about the horizontal axis as well. In that case, you would get a letter 'p' while the paper shows the letter 'b'. Now, isn't the mirror image now vertically inverted?

When we compare the real object and the mirror image, nothing dictates that we can't rotate the object about the horizontal axis to make it face us. The crux of this problem is in identifying that in our common experience, in order to make an object that originally faced forward face us, we would rotate it about the vertical axis.

Thus, the image of a plane mirror being laterally inverted is NOT a physical property of a plane mirror. It is due to the way you rotate the object to make it face yourself.