Needing a visa and needing a passport are two different things. Generally speaking, you need a passport when going abroad. Many countries do treat transit differently when it comes to visa requirements but there is certainly no general exception to the passport requirement.
That said, Turkey accepts ID cards for a handful of nationalities including Switzerland but not, as far as I can tell, Bulgaria. There is also no need to get a visa for transit if you don't need to leave the “transit lounge” of the airport and Bulgarian citizens can also enter Turkey without a visa.
Consequently, it seems you could travel only with your Swiss ID, while your wife should take her passport with her but will not need any extra paperwork or payment (visa or e-visa). Note that the airline will check this before boarding the plane in Switzerland so you can ask them what they require for your trip.