The Feather and the Mini is connected via two pins - Feather pin 10 to Mini pin 4, and Feather pin 11 to Mini pin 5. Their grounds are connected. This setup is for a very simple communication and it works fine. The Feather communicates to the Mini by setting either or both pins high while the Mini has its pins as INPUT_PULLUP.
Today when testing, the Feather was powered through USB but the Mini was left without power, but the pins still connected. When the feather at some point in the code set its pins high, the Mini turned on without having power through raw or USB. The only power it received was through those wires.
How can that happen? Will it damage either boards?
That happens all the time with improperly wired circuits and can damage one or both boards. You are powering one module through an input pin by the other.
Either power on/off both devices at the same time, or put a 10K resistor between the pins to limit the current. The latter approach can still lead to malfunction of one or both devices.
You must not apply an external voltage higher than Vcc or lower than Gnd to any of the I/O pins without current limiting (aiming to keep current under 1mA maximum). When the board is not powered, Vcc = 0, so any external voltage can cause a problem - the pins are protected with internal diodes to the power rails, but these are not meant to carry much current.
Backpowering the processor through it's protection diodes is an excellent way to blow the pins involved.