I once met a software engineer who said I was not full-stack because I didn't know how to add to the Linux Kernal. I once met a hiring manager who said I'm not front-end because I don't know how to vertically shard MySQL and "every front-end developer knows how to do that." Technology stacks differ per company and the definition of the title will also differ. That doesn't mean it doesn't exist.
I consider myself a “full-stack” programming language engineer because I’m equally comfortable with type theory, user experience design, practical implementation, and instruction-level optimisation. Am I a full-stack web developer because I’m equally comfortable with front-end design & development, server-side infrastructure, and networking protocols?
Some would say yes, some no—but I can readily develop, maintain, and optimise a whole application by myself if I need to. I may not do as good a job in any particular narrow area as a specialist in that area, of course.