His approach is a bit different as he is shimming 16 bit APIs to their 64-bit equivalent. Wine reimplemented them to run on top of a POSIX environment.
His approach will result in higher fidelity for apps as they are basically running on Win64, just with a 16-bit virtual machine driving it.
Wine apps running on Windows would likely feel non-native as the UI components are reimplemented from scratch.
Wine actually has (mostly) functional theming support-- point Wine to a .msstyles theme DLL and it'll draw themed controls (rather than the Windows Classic style default).
They just don't ship a theme with Wine; understandable for licensing reasons (they can't distribute Microsoft's designs), but it means that Wine looks like Windows 2000 out of the box.
I find it a bit amusing that were they to "update" their look to that of the flat Windows 8+ UI, it would probably entail the removal of a lot of code.
...and yet even on a 25MHz 386, no one complained about the slow drawing speed of the "fancy" 3D buttons in Win3.x.
Well, modern flat design generally also includes a much more animated interface, which would probably entail the addition of much more code. And I don't think the drawing speed of buttons was ever a very strong reason for the rise of flat design - though I completely understand your distaste for it.
16-bit applications can't run in long mode (i.e. a 64-bit OS), so you'll need a 32-bit OS to run these apps in WINE. WINE is just a shim and a library reimplementation, not an emulator, so it just passes the instructions on to the CPU, and you're limited by what the CPU can do.
Hmm... I wonder if you can run 16-bit applications in WINE on a 32-bit VirtualBox VM running on a 64-bit Linux system. That sounds like overkill, though. Might as well just use PCem or DOSbox, depending on your use case (PCem: cycle-accurate emulation of a whole PC, including specific hardware models and running any OS a real PC could; DOSbox: playing some old games for fun).
TBH, I'm not sure what win3mu's point is, either: it probably won't be open-source, and right now the only reason to not use PCem or DOSbox is if you don't own a legitimate copy of Win3.1 and you're not willing to pirate it. An open-source clone of Windows 3.1 that can run any Win16 program in long mode (by interpreting the instructions like an actual emulator) would be a great idea, but this isn't it.
But yeah, I don't think it's explicitly supported since it's not their target market :) Codeweavers' focus is running Windows apps on Mac and Linux (Crossover - go support them! https://www.codeweavers.com/)
Edit: It looks like they completely dropped support for it as I can't find any of the old articles anymore; Seems it only supports mingw setup now. https://wiki.winehq.org/Cygwin_and_More
Crossover is a commercial version of Wine which includes a bunch of extra hacks for popular applications. It comes with support. It's how the Wine project pays salaries.
PlayOnLinux is a Wine wrapper which has preset configurations (best known working wine version and dll overrides) for a ton of popular games/software. Think of it like a virtualenv on steroids.
(In fact, it strikes me now that if you're on Windows 10, you could probably run Wine on the WSL.)