There really is no rule that states that..... just some general principals on living/working/free flow of capital/etc. in all members states.

All financial institutions are free (and encouraged) to have their own client acceptance policies. And since a lot of countries have quite strict regulations for those it's a lot easier to just accept residents because it takes a lot more effort (and in the case of US passport holders even more) to service non-residents. And it carries a higher price as well.

In reality there still is no single European Financial market for banking. What they are implementing now is "just" that payments between EU countries don't incur charges (some exceptions) and are executed more efficiently. For instance there is no credit agency that covers the EU....

The problem with Belgium bank accounts is that in general they are expensive to hold. Hardly anything like the DKB free cash account with a credit card.