I don't know, I would tend to side with the theory that the car engine sound at the specific RPM and power output matches the resonance frequency of one of the house windows. Resonance can creep up in many phenomena.

But the engine speed doesn't change until you lift the clutch. So if this were due to the RPM, putting it in reverse while stationary surely wouldn't do anything in particular.

If it’s an automatic, putting into reverse will change the engine RPM.

Fuel mixture changes as the engine heats up, changing the sound

Not in my opinion because this would have been a problem before with other cars and alarms. Or motorbikes.

Without knowing the details of the alarm system it is hard to have an educated guess.

I've owned a number of Subarus and let me tell you, even the non-performance models have a way of shaking things inside the house that you wouldn't believe until you experience it. My STI was capable of being loud, but oddly enough it was my wife's Outback (nearly silent if you stood next to it) that would shake pictures on the walls and make them start tapping.

Very true. My '13 WRX shakes pictures when I start it up in the garage, I have to immediately pull into the street so I don't piss anyone off while getting Google Maps and Spotify set up before I drive.

Motion detectors with microwave, especially residential sensors, don't use it as the sole detection method. It's always paired with passive infrared. Microwave interference doesn't really make any sense because he'd have to be tripping the infrared and microwave source at the same time.

Source: I wrote firmware for microwave/PIR motion detectors for a few years.

His WRX doesn't have backup sensors.

It could have backup sensors.

Rear cross detection and blind-spot detection is an option package.

This allows my car (2016 WRX Limited) to warn me about cars flying by when I am backing up out of a parking spot for example.

I have that on my Mazda 3 and it's absolutely brilliant. The range and sensitivity of that system is rather amazing.

FYI car makers are starting to collect data from connected cars using their ultrasonic sensors for collision avoidance, among other sensors. You can indeed use them to map surroundings fairly accurately.

Mine isn't connected but, yep, could certainly see how that'd work. Whilst no fancy auto pilot yet, the AEB and radar cruise are very nice things to have in a car.

I have 2014 3. This system has spoiled me. If my next car doesn't have something similar, I'll probably crash it pulling out of dealership.

Yep, same model / year as mine. 2.5l MT Astina. Great car!

The reverse lights triggering motion detection and engine/transmission vibration both seem plausible too.