Where's "here"?
Must be an interesting place. Personally, I've never seen a washing machine with an induction motor.
I have seen one, in an apartment building I lived in while studying, it was an old "industrial" machine, with three-phase supply. Likely has two windings, high-speed (likely 2 poles per phase) and low-speed (more poles). The obvious downsides of the induction motor, despite dual windings, is large size and lowish spinning speed, because the speed choices (for 50Hz) would be, for example, 600 and 3000 rpm. There was a separate machine for spinning the laundry afterwards, also used an induction motor but now geared for high RPM. That did a great job.
People often forget mains frequency driven induction motors are very crappy, practically useless for all other purposes except constant-speed near the rated speed. This feature also makes "starting" them awkward.
A VFD (inverter) fixes this fundamental problem. But a brushed motor does the same in a simple way (it's a mechanical "variable frequency drive"), and brushed motors can actually be very good if constructed well.
So being able to generate acceptable torque with acceptable efficiency
at any RPM is one reason to use brushed motors (against fixed frequency AC motors), but another one is miniaturization, they can run with higher internal frequency, higher RPM, and thus, less iron. This is specifically why they are used in vacuum cleaners, compare the 1kW vacuum cleaner universal motor with an 1kW induction motor, the weight difference is like 0.5kg vs. 5kg! The exact same thing why 400Hz AC motors are much smaller for the power than 50 or 60Hz motors.