My advisor told me that a Borel set can only be finite, countably infinite or having the cardinality of the continuum (obviously we are not assuming Continuum Hypothesis). I think he mentioned "Hausdorff-Kuratowkski Theorem" but I can't find that theorem nor any related. Do you know something about that? Since we were speaking about topological groups, more specifically abelian, Hausdorff, compact and metrisable, maybe these hypothesis are required.
Note: if I won't be able to find an answer in this site I will return to the advisor asking for a more precise statement and I will answer my own question. But what I hope is not to having to return back to the advisor and asking again the same question (the one in the title).
BIG NOTE: I am not asking for the cardinality of the set of all Borel set.
An even stronger result, due I believe to Suslin, is true: if $X$ is an infinite analytic subset of a Polish space, then either $|X|=\omega$, or $|X|=2^\omega$. This is Corollary $2$C.$3$ in Yiannis N. Moschovakis, Descriptive Set Theory, Second Edition, $2009$, freely available here [PDF]. The result for Borel subsets of Polish spaces can be found in David Marker’s lecture notes Descriptive Set Theory, available here, as Theorem $\mathbf{2.25}$; the extension to analytic sets is Theorem $\mathbf{4.16}$.