Replacing the thermocouple for a multimeter?
In general any thermocouple can be used with any meter that handles thermocouples as long as they are compatible.  That means that a K type thermocouple must be paired with a meter that is calibrated for K type thermocouples. K type thermocouples are the most common so it is very likely that your meter can handle the one you looked at on Ebay which is a K type. However there are much cheaper K types available than that one which is specified for very high tempertures. As far as calibration goes, there are standard tables of thermocouple voltage versus temperature for each type of thermocouple. This is possible because each type of thermocouple uses the same composition of wires.  Thus your multimeter would have been designed to use the table for K type thermocouples.  The actual accuracy you can achieve is determined by the specified accuracy of the thermocouple and how well the meter conforms to the standard table values which are nonlinear.