Indeed, that seems to be the reason. I used a scope to measure the frequency of both hybrid transponders, the one that works is exactly on 5MHz, the one that doesn't is closer to 5.03MHz. Also its signal is a bit distorted (looks a little bit like a sawtooth).
It's quite an old transponder, and surprisingly its frequency is on the high side rather than low, I double checked that.
I'll try the fix and try to save the signal of both working transponders The ghost transponder numbers are always the same, ie each of my two working transponder appears as 2 different IDs : its true ID, and sometimes a second "fake" ID as well.