Our previous washing machine had a mechanical rotating switch, sort of like an egg timer, built into it. I seriously thought about just gluing a pair of metal bits onto it to make a physical connection when it was done, which would either do something clever like trigger a RasPi into sending me a text, or something stupid like physically triggering a doorbell chime.
Can confirm this is super easy. It has the additional advantage of monitoring power usage and it allows you to cut power if the leak sensor under it goes off
I would go for the "monitor a number" before "sound recognition", unless you're talking about just using an amplifier to bring the chime into the entire house.
I'd dump cut the mic and wire it to some input before actually trying to check for the sound. But it's probably inconvenient to get to and monitoring overall power draw would be easy.
Does your machine run on petrol or manual labor? In that case, listen for an engine or someone complaining.
I'm certain 99% of washing machines are powered by electricity, and if you live in 120V town and need 3 phase you can monitor power using the magnetic field on one of 3 conductors.