Flies like an emulator, quacks like an emulator, waddles like an emulator: duck-typing says it's a bloody emulator.
The word "emulator" in tech sometimes acquired an association with CPU or computer emulators. So for whatever reason (joke, marketing, etc.) WINE changed the meaning of the acronym. But it's still an emulator, just an API emulator rather than CPU emulator.
It's likely they did it for the same reason for which "Windows Commander" famously had to change its name to "Total Commander": to avoid infringing upon Microsoft's trademark. Christian Ghisler got a letter from Microsoft's lawyers about the name of his file manager, though [1]; AFAIK, Wine didn't.
"The phrase "Wine Is Not an Emulator" is a reference to the fact that no processor code execution emulation occurs when running a Windows app under Wine. "Emulation" usually refers to the execution of compiled code intended for one processor (say, x86) by interpreting/recompiling software running on a different processor (say, PowerPC). " : ... In Wine, the Windows app's compiled x86 code runs at full native speed on the computer's x86 processor, just as it does when running under Windows. And Windows API calls and services also are not emulated, but rather substituted with Linux equivalents that are compiled for x86 and run at full, native speed."[1]
Except for the Windows API, including all of its bugs and oddities.
Emulation != Vitualization
Back in the day, us of the emulation community would refer to such things as "wrappers" (see, glide[1]). WINE definitely qualifies as a wrapper, as it merely translates one API to another. Darling seems to fit this same definition. Both are handling big API's, but that doesn't make it emulation when you reimplement them.
Wikipedia's definition mentions hardware specifically as a requirement of Emulation [2]. In fact, they go even further to say that any software that duplicates the hardware of another machine should be considered a simulator [3]. This means that WINE and Darling are now twice removed from the Emulation family.
[1] http://www.glideunderground.com/modules.php?op=modload&name=...
[2] https://en.wikipedia.org/wiki/Emulator
[3] https://en.wikipedia.org/wiki/Emulator#Emulation_versus_simu...