When birds make nests outside of their base anatomical functions, are they "unnatural"?
To be fair, "unnatural" is sometimes extended to refer to "life" instead of "humans", typically only when talking about the evolution of life or the search for extraterrestrial life. In that sense, then, yes - bird nests or termite mounds or coral reefs are unnatural. A more common wording for this same idea is something like "not created by geological processes".
What gives humans the special designation of their byproducts being unnatural?
You can argue that "unnatural" was a bad choice for this concept. But that's irrelevant to what words mean. There are other words like this - for example, "antisemitic" means "something that is against Jewish people", even though "semitic" means "of Jewish or Arabic or Phoenician etc. descent". So something discriminatory against semitic peoples is not necessarily anti-semitic.
Natural language doesn't follow strict logical rules. And of course, natural language is in fact itself an artificial, unnatural, construct.
It can both be true that everything humans do or create is natural because all of existence and everything in it is natural, and also that some things humans do or create are artificial or un-natural, without contradiction, because context may be taken into account, and the same words can mean different things depending on how and why they're employed.
Did life begin with the geological process of protein chains forming in geothermal vents? I don't know, but it begs whether it is natural or not.