Applying to grad school where I've been on exchange I know that many universities in the US and an increasing number in the EU are against taking on their own graduates. Does the same apply to exchange students? I'm wondering if I didn't make a strategic mistake by going there. On the other hand, I did everything to go on exchange to that school because I think they do great research.
I know that many universities in the US and an increasing number in
  the EU are against taking on their own graduates


I assume you mean for grad-school ? This is the first I've heard of it. We encourage our undergrads to search widely, but I'd love to take my own undergrads for grad school if I could persuade them to stay. 

The sentiment you're referring to is more prevalent at the faculty hiring level: namely, universities are less likely to hire their own Ph.D students as faculty (at least not without some time spent elsewhere first)