No.
Banks compete for deposits and compete to lend. The spread between those rates must pay the bank’s costs, cover its risks and turn a profit. When central banks reduce the return on safe assets banks hold, their income from lending to the government goes down. The banks thus seek to replace that income with other lending. Unfortunately, everyone in the system is doing the same thing. This competition forces lending rates down.
Savers are competing for investments next to the banks. There are lots of them crowding into bank deposits; this lets those banks lower the rate they pay. (Nobody forces savers to deposit Euros only in German banks.)
This is a system of feedback loops. Nobody is charging a tax, though banks may charge a fee for a service it costs them to provide. Depositors are free to take their business elsewhere. And entrepreneurs who think they can do better are free to launch competing financial services.