Thermocouples have stated tolerances [wikipedia]. However, the characteristic behaviour is defined by the materials in contact, so there aren't size or shape considerations as for RTDs. What are the main sources of the variation/tolerances in thermocouples?
A comment to this answer blames variation in metallurgical properties, which seems plausible, but this is unsourced.
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.
Thermocouples are nothing but two (or more) metals or alloys in contact, so the inherent variations must be metallurgical in nature. Metallurgical effects include the alloys, how evenly distributed they are during the drawing of the wire, and effects such as annealing and cold-working.
Of course the construction of a thermocouple can lead to differing measurements depending on way the thermocouple is made or mounted and how heat flows from the object being measured down (or up) the leads.
Extension wire is not always made from the same type of metals, or made to the same specifications, so the entire thermocouple assembly may show errors if the transition to extension leadwire is far from room temperature. The main reason for using different materials is to save money and allow the use of robust conductors, especially with precious metal thermocouple types (most commonly types S, R, B but gold has been used as well).
To understand thermocouples there are three things you have to keep in mind:
The most common source of variation in thermocouples is the wire (which results in changes of thermal mass and electrical resistance and in millivolt systems like thermocouples resistance is very significant). but many times significant source of differences in thermocouple readings is not the thermocouple, but the rest of the system. Common mode noise can be a problem as can rounding error. Wire length is almost never a source of error (assuming you are using thermocouple wire), but shielding, connectors and oxidation can all be problems, specifically hot junction oxidation (even on welded junctions) can lead to errors and is the most common reason to replace old thermocouples (Heat, dissimilar metals and current is a recipe for unwanted reactions.).
