@RobertBryant You are right, I just edited it. Thank you!

I have one last doubt. What happens if I consider $A=(f(r)/r)e_\phi$, but with $f$ not going to 1 at infinity?

Got it! Thank you again!

Thank you so much for the explanation! I think I understood the point. I don't know if we could use thin vortices here since $A$ is a 1-form, so it's supposed to have smooth coefficients (or else they could just be considered as long as the integral is finite), anyway your point on the thick vortices convinced me. My only question is: how do we know that the forms of the form $A=d\phi f(r)/r$, with $f\to1$ at $\infty$ (and maybe such that $f$ cancels the singularity at zero), are the only objects that $n$ is counting?

@AndreaBlass You're right, my mistake.