Transistors are simple electronical devices. They don't run software. You can control their inputs with another device (such a microcontroller) that does run software. You can also control their inputs with a button. You can't control their output with software.
I don't know how an Amazon echo is wired up, but if you just have a button connected to the gate of the transistor, it works basically the same as a mechanical switch.
Transistors have no registers. They have no arithmetic logical units. They have nothing. They are so simple they can be made up of less than 100 atoms. Transistors have to be connected electrically to other device. Any reverse engineer can trace what it is connected to and it's behaviour cannot be programmed. If you know that it's a transistor and you know the inputs, you can know the output. The same cannot be said for a device which runs software, you'd have to additionally know what that software does, which is incredibly more complicated.
Software is ran by microcontrollers. Transistors can be connected to microcontrollers. But they can also be connected to buttons. If there is no microcontroller, there is no software.
It's about as likely that the transistor is attached to a pin that sends an interrupt to the processor and it then applies a soft mute as it is the transistor is attached to a flip flop or register that toggles the mic getting power physically.
My guess would be it's controlled by software rather than directly by the hardware because then they can do whatever they want with the button via firmware or software updates. This includes nefarious stuff like a fake mute mode, or more innocent stuff like special behaviour on a long press vs short press.