I didn't do Erasmus, but I won a similar exchange program that brought me from the University of Granada to UC Irvine, where I met my now wife. Life changing experience.

The problem is that US universities charge exorbitant fees, and that creates a problem for exchange programs with (continental) European universities. American universities are mostly unwilling to waive fees for European students who are used to not paying any significant fees, and European universities are not going to pay for it on behalf of the students.

There's the Fulbright program with bilateral exchanges with many countries. According to wikipedia.de, on average each German university or college sends 55 students in a year to the US with a Fullbright scholarship.

May be better to exchange students between city and rural schools.

AFAIR the US already partecipates in the Erasmus Mundus program