It doesn’t seem like there is a tag slaughterhouses. There is the generic landuse=industrial + industrial=slaughterhouseTag:industrial=slaughterhouse - OpenStreetMap Wiki, but that’s for the generic area, not a specific business.
The key difference is that whole herds of live animals are brought to a slaughterhouse, while a hunter brings a single deer carcass at a time (or maybe a few) to a processing facility. Some may be large and might make sense to tag as industrial=* (or man_made=works). Others are small home based businesses that only operate around hunting season. These might more feel more appropriate under the craft=* key.
A deer processor can be accurately described as a “butcher”, but not a “butcher shop” because they don’t sell the meat. They just provide a service to hunters and give the meat back to them. So shop=butcher is wrong for this, but perhaps craft=butcher? It does have a little bit of use so far.
FWIW the 39 craft=butcher are here. Most are in Germany. A couple of the ones outside of Germany have indicated “not a shop” (also here). Some of the ones inside Germany also look “larger than just a shop”.
Notwithstanding that, a deliberately different tag (e.g. as suggested above) would avoid any confusion.
In my local neck of the woods the hunters process the game close to where they shot it. They do that themselves, or if they are of the kind “wealthy shooters buying a shoot” have rented hands to do that. Anything else will require a butcher and quick moving of the body.
UPDATE to clarify: Hunters just remove the intestines, cutting the meat most always with a butcher.
(OT on slaughterhouses) I suspect that many just aren’t mapped at all, because they often don’t advertise their presence much (thought locals will of course know that they are there). FWIW in the UK there’s a list here (under OGL) and the nearest one to me (actually part of the Food Standards Agency) is in the unmapped pink space here.