Thermocouples create a measurable voltage difference given a thermal energy difference*. This is the Seebeck effect. A "K-Type" thermocouple uses a particular alloy combination which is, as the Wikipedia article states, a general purpose design that varies widely because technology has advanced since its inception.
"...a wide variety of probes are available in its −200 °C to +1350 °C ... range."
Therefore, K-type alloys can measure up to 1350°C but not all products are designed with this upper limit in mind.
As Nick pointed out in a comment, you can find thermocouples designed for high temperature. For example, the Super OMEGACLAD® K-type have nearly such tolerances.
What makes this different from what you found on Amazon is the build quality and materials for the probe sheathing (high temperature probes use mineral insulation), conductors (not the chromel/alumel alloy, but the electrical connections to it), and probe length.
Consider an analogy. Let's say you found a very basic definition of automobile:
"Automobiles are transportation machines which can travel up to 429 km/h."
While this may be true, an inexpensive Honda Civic tops out at 209 km/h (ref). You'd need a Bugatti Veyron Super Sport to reach that upper limit (ref). What's the difference between a Honda Civic and a Bugatti Veyron Super Sport (other than about 2.4 million US dollars)? They both use combustion engine technology, but with radically different quality components.
* EEVBlog has a basic video tutorial on how thermocouples work.