Yes, the word "thing" has many meanings. For example, the German dictionary by the Brothers Grimm lists the following Latin terms as possible meanings of "Ding" (= "thing"): ens, res, substantia, aliquid, causa, judicium, forum, conventus, status, opes, persona.
Whether it is true that everything is not a thing depends on which meaning is used, and it also depends on the chosen domain of quantification of the universal quantifier "everything": Does "everything" refer only to existent things, or to both existent things and nonexistent ones. (In the former case, "Everything exists" is necessarily true; and in the latter case it is false.)
* If "thing" means "something" ("aliquid"), then "Everything is not a thing" means "Everything is not something/anything" = "Everything is nothing". What does "being nothing" mean? It means "being identical to nothing" or "having no essence/nature/Sosein".
Every existent thing is surely identical to something, and it also has some essence/nature/Sosein. Whether this is true of every nonexistent thing as well is a contentious issue among philosophers. Alexius Meinong is famous for having argued that a thing's Sosein (essence) is independent of its Dasein (existence), and there are philosophers who argue that nonexistent things are still self-identical.
* If thing means "entity" ("ens"), then it is false that everything is not a thing, because all existent things (at least) are entities of some kind or other.
* If "thing" means "res" in the narrow ontological sense of substance ("substantia") or object, then "Everything is not a thing" means "Everything is not an object/substance" = "Everything belongs to some ontological category other than <object> or <substance>". Whether this is true is a contentious issue among ontologists.