multimeter temperature accuracy
Let's take a typical midrange handheld multimeter manual so we know what the heck we are talking about: 

The relevant section is here: 



Accuracy is stated as +/-3% of reading + 3 digits. That means if your temperature is indicated at 20°C, the multimeter is between 18°C and 28°C, a perfect thermocouple junction would be between 16.4°C and 23.6°C. 

In practice that might be a bit optimistic, especially for Fahrenheit, as it should include linearity errors as well as reference errors and cold-junction compensation errors, and we would not expect those to improve for °F. It will also likely be in greater error if there are large temperature gradients in the vicinity or if temperature is changing rapidly with time. 

Typically the reference and linearity errors are more of an issue at very high temperatures. And, of course, the sensor errors, which are not included in the multimeter accuracy. Conservation of energy and thermodynamics limit the sensor errors near room temperature, of course. 

Turning our attention to the sensor, limits of error for a Chromel-Alumel thermocouple are as follows: 


  Type K Accuracy (whichever is greater-- over the range 0 to 1250°C):
  
  Standard: +/- 2.2C or +/- .75%
  
  Special Limits of Error: +/- 1.1C or 0.4%


That's for standards-compliant properly manufactured thermocouple alloys. 

You can improve the accuracy, especially of measurements near room temperature, by immersing your probe deeply into a ice-water slurry, taking a 0°C reading and correcting your actual reading for any error you see. For example, if your meter reads -1°C in the slurry, add 1°C to your reading. And, of course, do it all in a stable and temperature controlled environment as free of drafts as possible. That will correct for cold-junction compensation and any small thermal EMF errors which dominate near room temperature.