The temperature sensor and the wires are one in the thermocouple.
The temperature must be known where the thermocouple meets the circuit board. This spot is called cold junction.
The DIN EN 60584 standard specifies the thermal voltage to be expected in relation to 0 ° C. This is important to note because the thermal voltage caused by the Seebeck effect is not linear.
In most cases, an RTD (PT1000) is connected to the connector on the cold junction in a thermally conductive manner. The measurement inaccuracies from the thermocouple and from the RTD thus flow into the temperature measurement. For this reason alone, temperature measurement with thermocouples is complex and less accurate than just an e.g., PT 1000.
The length and the temperature gradient along the sensor cable also play a role. If you have many measuring points to monitor, the length of the extension cables should be the same to ensure better comparability. To ensure this, the individual measuring lines are lengthened to the required length with extension cables or compensating cables.
So yes, you can use audio connectors if you can take the temperature of the connector into account. Still, I don't think this should be done because these connectors don't evenly distribute heat well. Sockets for thermocouples have better properties in this regard and are also more suitable than anything else in terms of their mass.