can two separate circuits use the one sensor
Generally speaking, you would need to figure out how the washing machine is using the thermistor and then you might be able to use the same sensor by adapting your circuit. In the best case, it could be as simple as a voltage divider that could fairly easily be adapted, or it could be some AC scheme as is used in consumer thermometers. If the excitation is brief and periodic, it could be difficult to use it.
Also, thermistors have a wide variety of possible characteristics (not just the resistance at a reference temperature), so an unknown thermistor may not be all that easy to use accurately, even if the circuit issue is solved.
Caution- if the circuitry in the washing machine is not isolated from the mains, some very bad and potentially lethal and/or very expensive things could happen.
If you're not up for the reverse engineering  you might find it easier to just add a second sensor of known characteristics.