Do I have to register with the Federal Aviation Office as the operator of the drone?

Yes. As a tenant of the drone, you are considered to be the operator of the drone. According to EU drone law, operators of drones must register with the Federal Aviation Office (www.lba.de). You can find answers to frequently asked questions about registration at the Federal Aviation Office at:
https://www.lba.de/DE/Betrieb/Unbemannte_Luftfahrtsysteme/FAQ/Uebersicht_FAQ_node.html.