Does the saying "Allah is between me and you" imply prohibition, and is overstepping it considered disbelief, and what are the consequences of that?

This statement means refraining from speaking, in adherence to God’s prohibition, and it does not lead to disbelief (kufr). The most that can be said is that it functions as an oath if the person intended to forbid themselves from speaking. In that case, expiation (kaffarah) becomes obligatory according to those who consider such a prohibition an oath, bearing in mind the scholarly disagreement regarding the obligation of expiation for one who forbids something upon themselves. It has been shown that this phrasing does not function as an oath and nothing is obligatory because of it.