Sadly the rules are, if not silent, very quiet on this issue. In 3.5e this was handled by the Extraplanar subtype that creatures acquired when they traveled between planes and the rule that any creature that was not Extraplanar defaulted to being a native of the material plane. 5e is a brave new world though so we can't just use those rules.
Native, or indigenous, species are species that exist in a given ecosystem without being introduced by human activity. This makes sense in our world where the term is generally used when distinguishing between parts of an ecosystem disrupted by human presence and parts that we haven't gotten to yet.
So how about D&D? In D&D the term, "native to this plane of existence", could mean, "is meant to be on this plane of existence and is only found elsewhere due to intelligent intervention". Let's look at how that works out.
- Celestials, Demons, Devils, all have their own planes to be on even though they often leave them.
- Elementals the MM specifically says they are native to the elemental planes and that matches our definition since they only leave if someone summons them.
- Aarakocra are interesting as despite being humanoid they are indigenous to the plane of air. They also exist on the material plane, but it is mentioned that they still hang out near portals to the plane of air.
- Slaads are an interesting case because while they live in Limbo they need humanoid hosts to reproduce and those are in short supply there. I'd say Slaad are actually native to both the material plane and limbo since they appear to commonly use both for their life cycle.
- A human shape-shifted into a fire elemental is still a human for our purposes and so native to the material plane. Sure they look like a fire elemental, have the powers of a fire elemental, and are near indistinguishable from a fire elemental, but it's only due to (hopefully) intelligent intervention that they got that way.
Using this definition leads to some tricky questions but generally works. Importantly it seems to match the intention of the spell to send creatures back where they belong.