No, this isn't related to chlorine being stabilized. Stabilized chlorine in a pool means there is Cyanuric Acid (CYA) there and the purpose of it is to lessen the degradation of chlorine from sunlight. What is happening with chlorinating liquid or bleach isn't related to sunlight -- exposure to sunlight would have it degrade VERY quickly. Instead, the issue is one of high concentration (in liquid form) and temperature. Every doubling of concentration increases the degradation rate by a factor of 3 or 4. Every 10ºF increase in temperature roughly doubles the degradation rate.
Solid forms of chlorine, such as Trichlor, Dichlor, and Cal-Hypo, degrade much more slowly mostly because they are kept dry, but they do degrade as a function of temperature. Nevertheless, such products typically last for a year or more unless stored in hotter environments.