As long as you can relate your proficient Intelligence or Wisdom check to your terrain, you can double the proficiency bonus.
It is not far fetched or fanciful to use your knowledge of herbal remedies available in the area, sources of clean water, natural anti-biotics etc to benefit a Medicine check to treat an injured person in that terrain.
In fact the Ranger/ survival expert knowing local remedies, healing herbs etc to treat an injury is so common, it's virtually a trope.
I mean, Aragorn himself did this in LoTR.
I would have no issue with it at all. It goes to the RAI, doesn't break RAW and importantly also goes to the RAF.
Two things, covering them in order of importance.
1) You have completely changed my entire point. I was going along with it, but I need to stop right here and remind you what the original point was.
IDENTIFYING THE INJURY
That does not use knowledge of local remedies, healing herbs or clean sources of water. That, in fact, does not deal with treating the wound at all, except that you have a much harder time treating a wound if you don't know the source.
2) If, because I know the forests, I know that Deepwood moss as anti-venom properties that makes it perfect for dealing with things like giant spider bites.... why can't I have a stock of Deepwood moss that I can use at any time to get expertise on the Wisdom check? Honestly, since I have to have proficiency in medicine anyways to get the expertise, wouldn't I know of other, similar natural remedies? Like that Frostflowers of the tundra have similiar effects to Deepwood Moss? I am trained in medical knowledge, and knowing which remedies can be substituted sounds like something I'd know.
And heck, this is all just from knowing that the plants exist, what if I had proficiency in Nature and a Herbalism kit, shouldn't anyone be able to get expertise on the medicine check then? I'm an expert on plants, where they are in the natural world, and how to apply them.
And so, we end up circling back. Why can I only get this expertise when it directly involves mountains, when that seems to be the least important part of the requirement. And, I know quite a few DMs who would not allow this. You don't get expertise on treating this wound just because you are in the mountains, because the wound itself is not something related to the mountains, and you as the player don't get to add in useful medical herbs nearby your location. Because the moment you do, then you can just go around and collect those herbs, and get expertise on every medical check. Which they would say breaks RAI.