A timeline:
Intel released a bunch of CPU architectures, naming each one after the CPU that it originated with (8008, 8080, 8086, 80186, 80286, etc). These, collectively, became known as 'x86', as each new version was backwards-compatible with the previous. These were 4-, 8-, and eventually 16-bit.
Intel released their 32-bit extension to their x86 instruction set, and named it 'i386'. Also known as IA32 (Intel Architecture 32-bit).
FreeBSD Project ported FreeBSD to that architecture, and named it 'i386'.
Intel released a new 64-bit architecture that they named IA64 (aka Itanium), expecting everyone to migrate to that from i386.
FreeBSD Project ported FreeBSD over to that architecture, and named it 'ia64'.
AMD released their 64-bit extension to Intel x86 instruction set, and named it 'amd64'.
FreeBSD Project ported FreeBSD over to that architecture, and named it 'amd64'.
Intel realised AMD had them over a barrel as no one sane had Itanium systems under their desk and AthlonMP/Opterons were destroying P4/Xeon systems, so they licensed the 'amd64' extensions, screwed up a bunch of things, made things just slightly incompatible, and named it EM64T. Then renamed it IA32e. Then renamed it Intel64. Then rename it something else. And, I believe even renamed it once more after that. No idea what it's actually called nowadays.
FreeBSD Project 'ported' FreeBSD to Intel's version of 'amd64'. Since only minor changes were needed, and it's not really a separate CPU architecture, it's name has not been changed. The 'amd64' version works on all 64-bit, x86-compatible systems. Even ones from VIA.
Along the way, a bunch of Linux devs decided that 'amd64' was confusing and decided to completely remove AMD's historical and ground-breaking contribution to x86 computing, renaming their CPU architectures to 'x86' and 'x86_64' (which really rolls off the tongue). Somewhere in there was also a suggestion for 'x86-64'.
Not to be out-done, several more people online decided to completely remove Intel from the equation as well, and started referencing the 64-bit architecture as 'x64'. Meaning the 'new and improved' architecture is 12 versions older than the ancient architecture (somehow x64 > x86). I believe Microsoft is part of this group.
Thankfully, the FreeBSD devs have not fallen into the 'nom-du-jour' games of the others, and continue to refer to the CPU architectures by their original names: i386 and amd64.