What is the main source of variation between thermocouples of the same type
In my experience the difference between thermocouples of the same type typically comes from the the manufacturing process. I have been making a number of K-type thermocouples using electrical arc welding, with "exactly" the same wire lengths, at the same temperature and with the same amplifier they display different temperatures in the range of ±0.5 Celsius from each other. 

My only explanations for this is that during the fusing different amounts of impurities are introduced into the junction, which is mostly based on the quality of the conductors, and that the phenomenon is related to the temperature of the junction at the moment of fusing. 

Unsurprisingly Wikipedia has some useful info on this as well.

Basically, I'd say the guy in the thread you linked is right.