What kind of motor is this? Looks more like a solenoid than a motor
It is a motor
A shame it's toast.  I found a couple of places selling them - over $80US in both places.
It's a stepper motor, but apparently no longer in production - the manufacturer doesn't list it anymore.  Epoch homepage.
At a guess, I'd say yours is much like the others listed on the Epoch page:  Two phase bipolar stepper.

As to how it can be a stepper motor, it would appear to be common to have the two winding coils of a bipolar stepper in a kind of &quot;stacked pancake&quot; setup.
This page shows the construction of a bipolar two phase stepper motor:

The windings clearly run around the motor like the ones in your motor.

The second photo you've added shows how it works.
The rotor will have &quot;stripes&quot; of north and south poles around it.  Like you'd put a bunch of bar magnets in a cicle, north ends to south ends around the circle.
These &quot;stripes&quot; will be about as wide as the &quot;teeth&quot; around the inside of each coil.
You have just two coils, but many pole pieces against many magnets.  Each pulse will only rotate the motor the width of one &quot;tooth.&quot;