> C major is identical to A minor, and I don’t understand why we need both.

Well, an orange and a carrot are the same color, but does that mean we don't need both? These two keys have the same signature, and that's it. To say they are identical is false.

> I don’t know anything about music.

You got that right.

> I suppose it’s possible to change the sound of an entire piece of music just by changing the key signature, but does anyone actually do that?

Only about every major composer who has ever lived. Look through any Beethoven sonata and see all the key signature changes throughout.

Your article didn't even hit one of the most important parts of music, modulation, which if you understood it would make many of your other confusions go away.