TL;DR: While Intel had their heads shoved up their ass making the Itanium architecture, AMD made a 64-bit variant of x86 that was backward compatible with the older x86 ISA. Technology moved on, and amd64 was adopted while Intel kept trying and failing to push their binary-incompatible architecture.
Eventually, Intel had to give up and adopt AMD’s amd64 ISA. In exchange for letting them use it, Intel lets AMD use the older x86 ISA.