I found a solution myself. The modules are working as they are supposed to and are not faulty. Also I did nothing wrong.
The Problem are my development machines. Mac mini (Late 2009) and MacBookPro (Mid 2009), both having a Broadcom 2046 Bluetooth host chip. I used those to connect via Bluetooth and open a tty session using minicom. Running macOS and using "Hardware IO Tools" to log Bluetooth packets I saw that there was clearly nothing coming in.
After that I turned over to my Windows and used Hype! Terminal. Everything worked!
Then I went to my Raspberry Pi and Linux. After pairing using bluetoothctl and issuing a few commands from Bluetooth as ttyUSBBluetooth as ttyUSB. Everything worked as well.
Then I tried BlueSPP on an Andriod phone. And again, it worked as it was supposed to.
So my conclusion is that there is some sort of incompatibility between the SPP-C module and the Broadcom 2046 host controller.