Godot is a game engine. I'm not a game developer myself but I expect that "full stack" in the context of game dev to refer to a developer who is able to code all parts of the game: physics, graphics, level design, interactions and so on. If you really want to learn all those things (I mean learn them at their core) then Godot might not be the right approach since it already offers a lot of things to you, so that you won't have to implement those yourself. If gamedev is your focus, then I'd suggest going for something much more low-level such as libsdl. Check it out and see what you think.

On the other hand you mentioned Vue, Django, PostgreSQL and Apache. Vue, Django and Apache are web frameworks/technologies. They are used to build websites. This is very different from game development which is a bit confusing to me because I can't really understand what you want :). If you want to go into web development then yes, those 3 are great for doing that.

I specifically left out PostgreSQL because this is the kind of tool that you will use regardless of the path you choose in your career. Websites use databases, mobile apps use databases, games use databases, everything in the world of software uses a database at some point. So this is a skill that you really have to learn anyway. Note that there are multiple such tools (MySQL, PostgreSQL, SQLite etc). I haven't personally used PostgreSQL but I've read some pretty good reviews of it online so I think this is fine for you to start using and learning.