They did exist the same way people and goods did travel across Eurasia, but it wasn't some defined political or economical entity and they did not call themselves "the third world"

Unlike "the Silk Road" or "the Middle Ages", "third world" was a term in contemporary use at the time it applies to, including among non-aligned states and NGOs that worked on third world cooperation.

It was pretty well-defined as political classifications go, and people involved in actual "entities" related to it were aware of and sometimes used the term.