We have a national student eID and at the same time our higher education institutions issue an extra eID even sometimes at faculty level. What eID shall be used for the European Student Identifier?

It is up to each country/institution to decide which student eID they use to issue the European Student Identifier. In the event you decide to use the national student eID, the European Student Identifier format is simplified since no SCHAC code will be required to guarantee the uniqueness of the eID. In the event you decide to use an eID that is issued at faculty level, you need to be able to guarantee that the eID is unique and persistent at the level of your institution since otherwise two students might get the same identifier. The specifications of the European Student Identifier are available here.