Not really. It is just translating the Windows system API calls into Linux system API calls. It's not emulating Windows, it's an entirely different implementation that doesn't necessarily match that of Microsoft's implementation. It had it own workarounds to make buggy code work.
You wouldn't call a Java Virtual Machine an emulator of another JVM either, they're just different implementations of the same specification.
Thing is, I do kind of think of a JVM as an emulator for a processor that doesn't exist.
WINE kind of blurs the line of a traditional emulator by having the executable run natively on the target machine's CPU, but everything it does in regards to dealing with the host OS, the display, disk access, etc, is emulated as far as I'm aware.
A theoretical PS4 or Xbox One emulator running on x86 hardware could be just as much of an emulator as WINE is.
Yes but an emulator emulates both the CPU and GPU of the consoles and in the case of PS4 even thought the CPU is x86 the biggest difference I can think of is the GPU drivers.
Maybe depending on how far you take it. A CPU instruction is different from hardware to hardware, but a function signature would stay the same no matter the underlying architecture. If we want to go through that logic then an interpreter can be thought of as a form of emulator.