@Philipp: Under the assumption the OP gave that the field is infinite, your counterexample does not hold.

I have removed my previous comment, which contained an error, and added the relevant part as an answer.

@Gortaur: I believe that you need to give some limitations on the space (namely, perfect, to exclude the end cases of isolated points) as well consider some "trivial" cases and try to see if some information can be said on $G$ (for example, you might want to have an example that $X\subseteq Dom(\bar G)\cup Rng(\bar G)$, or something else added). After you have investigated some basic (or even common) cases you can try to see how the general case holds.

@Gortaur: If they are given, please specify them. If they are arbitrarily fixed prior to the construction, what makes it impossible that in a non-perfect space $G$ and $A$ will be set as I commented above?

Gortaur: Considering the non-perfect case, if $A$ is a set of finitely many isolated points, and $G=A\times A$ then $G=\bar G$ and the condition holds. You might want to specify perfect polish spaces, not just any polish space.

@Konrad: Of course it carries over! It is very simple, a simple generalization of Blass construction (generalizing Lauchli) still yields a vector spaces without non-scalar automorphisms, and from here the proof by Todd carries over completely.

@Konrad: Something that I am working on nowadays shows that in Lauchli models it is consistent to have $DC_\kappa$ for arbitrarily high $\kappa$. I'm not sure that Todd's proof will carry though.

I just have to say that personally I have never found category theory to help me understand, remember or do mathematics. Did I mention that I'm in the field of set theory? I guess I didn't... Your post somehow diminishes set theory outside of mathematics, this is at least how I read it (even if it is not what you intended to write).

@David Speyer: This is a good point, I'll restate my question.

I must say that I have never in my studies heard any of my teachers speak of this "One universe of sets!", if it was mentioned at all (depending on the course of course) then it was mentioned as "a universe of sets" and nothing more.

Ricky: Wonderful answer. I have added some more to the original question.