So, the funny thing about this is that they're technically right? At least, sorta. My point is that it's not intended as a contradictory, incorrect name.
So an emulator is software that acts like some hardware: an NES emulator acts like the 6502 CPU in the NES and the graphics and sound chips and such.
The thing is, WINE is designed for x86 PCs, and windows is also (mostly: there's ports now) designed for those. And there's no special chips needed to run Windows that Linux computers don't have... So there's nothing to emulate.
So why can't you just run Windows programs on Linux without WINE? well, because of system calls.
Every x86 program (Windows or Linux) runs on the x86 CPU, doing the same sort of things, put they have to do syscalls to tell the OS to do things or get info from the OS. Things like "hey draw this message box" or "open a file" or "did the user push any keys?"
Those are implemented in the OS, and the program has to call out to them to do them. And the difference (to the program, at least) between Linux and Windows is how those syscalls are designed: Windows is designed around a graphical paradigm where you're expected to have a window in front of a user, and Linux built on a server model, where the computer running the program may not be the same one with the monitor connected, so they designed them differently.
So what does WINE do, if it has nothing to emulate? It's a translation layer. Basically, it implements the Windows API that windows programs use, but it implements it against the Linux API. So a program will call CreateFileW() (a windows API) and instead of Windows responding to it, WINE has an implementation of CreateFileW() that instead calls open(), which is the Linux equivalent.
So, in the technical sense, WINE is not an emulator. The problem is that it acts kinda like one from a user perspective, and you could totally use "emulator" in a non-technical sense and say that WINE is "emulating" the Windows API... but the name wasn't meant as a joke.
Also... WINE has changed their name. It's still "WINE" or "Wine", but it no longer stands for anything, it's just a name.