The plants evolved millions of years ago, before Israel or Lebanon existed. Given that it is mentioned in the Psalms, this spice seems to have been used by people long before the creation of the state of Lebanon. Furthermore, according to Wikipedia, other non-Arab peoples, like the Turks and Iranians, also use Za'atar. So, the Lebanese use Za'atar, good for them. But it does not make Za'atar Lebanese.
Opposition leader Isaac Herzog calls on government to allow Syrian refugees into Israel (Haaretz)
from the article: Tourist tip #248 / Za'atar, the ubiquitous spice