[RPG] To what extent may I customize a demiplane

Wiktionary: nature

2. The innate characteristics of a thing. What something will tend by its own constitution, to be or do. Distinct from what might be expected or intended.

5. Kind, sort; character; quality.

They’re using the word in one of these kinds of senses; no special game jargon. They’re just saying that if you’re familiar with what kind of plane the demiplane is, you can connect to it.

Connecting at random appears to be legit; one would presume that wizards storing valuables in their own private demiplane will be protecting that treasure in some fashion, not just relying on the demiplane’s limited privacy. That’d be security through obscurity, and wizards are too smart for that.

And yes, this is consistent with prior editions, which had more detail on demiplanes. Particularly in AD&D, where Planescape was a major setting, demiplanes got quite a bit of detail – and people snuck into demiplanes quite often.

Based on the ordinary English sense of the word "native", only creatures that actually originated in a demiplane are natives of that plane. In most cases (e.g., when the demiplane spell is used), the demiplane has no inhabitants, and thus no creature is native to that demiplane. Certainly there's no reason to conclude that the caster of a spell that creates or accesses a given demiplane is a "native" of that plane.

But this is a corner case, and individual DMs might have specific ideas about how they would like it to be handled. There's no real game balance issue here.