AMD invented 64-bit extensions back when Microsoft was busy trying to differentiate between their "big metal" Itanium chips and consumer processors. Turns out everyone wanted more memory and didn't want to have to rewrite all of their apps to get it.

Intel then had to license it, but because it's Intel they refused to call it AMD64. So the industry as a whole just started saying x86-64 to mean both AMD64 and Intel 64 so that the general consumer wouldn't think that insert app wasn't compatible with both.