Yes, this has worked right from the beginning. The correct ttyS0 and ttyS1 are offered in the context menu. ttyS1 is the one the 34401A is connected to. But TestController just reports ";; Physical Port S1 (ttyS1) Device HEWLETT-PACKARD,34401A, do not match: null" without the device going into REMote mode or anything, not even an error.
However, If I change the parity on the device intentionally from NONE (8 Data Bits) to EVEN (7 Data Bits), I can provoke ERROR 513 on the device when TestController tries to connect, followed again by TestController outputting ";; Physical Port S1 (ttyS1) Device HEWLETT-PACKARD,34401A, do not match: null".
So the connection seems possible, but still TestController does not seem to be able to retrieve anything.
EDIT: I will check the cable. I have the suspicion that receiving data from device is not possible at all, just sending commands to it.