Electronic – “duplicate” a stereo signal to 5 channels? Are there any impedance issues

Sounds like you should consider placing some good quality capacitors in series with the audio signal lines from the jack that you added and to where you connected the inputs to the guts of the old cassette player's amplifier input. I would start with a value of 10uF with a voltage rating of 16V or more. It would be best to utilize non-polarized capacitors but if you cannot find those then consider one of two options:

a) Use two 22uF capacitors in series wires + to + and then one - lead to the jack and the other - lead to the cassette input.

b) Try to see how a polarized cap would work. Put the + lead toward the cassette player amplifier and the - to the jack.

These capacitor ideas block the DC bias that is apparently present at the connection point where you attached the signal wires into the cassette amplifier board. Some of the earlier devices that you connected to the cassette jack may very well have had capacitor coupled outputs. On the other hand your newest device may not have an output configured like this.

I think, before over-complicating stuff with an op-amp, try this: -

It could be that the short circuit detection is using DC to decide whether it is shorted. The dc blocking caps shown are 10uF and in case these do not allow enough bass, be prepared to raise their value to 47uF.

You might also need to have bleed resistors from each input line to shield just to activate the output - try 10k on these.