Why are properties of components often given at 1 kHz?
Typically characteristics of components are given at some frequency that makes sense for the application and/or puts the performance in a good light. 

You are incorrect in giving a single frequency. Characteristics may be given at 120Hz, 1kHz, 10kHz, 100kHz, 1MHz, 1GHz or whatever. Electrolytic capacitors are often specified at 120Hz (full-wave rectified mains frequency) or 100kHz (in the same range as many switching power supplies). 

1kHz is a common frequency for specifying certain components (such as data converters) used in audio applications, probably because distortion at 1kHz will result in audible harmonics, whereas no human can discern an audible difference between a 10kHz sine wave and a 10kHz square wave. On the darker side, 1kHz is also high enough that certain types of distortion are less apparent- those caused by thermal effects, for example, though that's more of an issue with amplifiers than with data converters. 

Edit: In the case of ceramic disk capacitors, you've basically answered the question yourself - they state the test conditions including the frequency that their test instrument uses (1kHz), like your cap meter. Bias voltage is probably more important, and temperature for non-NP0 types. Note that loss is usually spec'd at a much higher frequency such as 1MHz.