I change carbon every two weeks. I rinse in tap, but all my faucets have undersink KDF filtration, so I guess it's not "tap."
Carbon does not discharge anything in the water until it has exhausted it's capacity to absorb, which is dependant on the pollution it is filtering, and the grade and type of carbon being used.
Water filtration systems with carbon are replaced according to the quality of water it is filtering and the amount of water processed through the filter, not by some arbitrary time period. Time periods are offered as a reference based on average use, so your typical layperson has an idea how often to replace it. LA water is NOT average use!
The way the carbon is packed and particle size also affects longevity. Filtering tap water for drinking is "light duty" for carbon, since it has already been filtered, and carbon used for this can often last for months, especially well-packed, high quality filters.
Filtering waste from a fish tank is heavy duty! How often YOU need to change your carbon will depend on the tank: it's inhabitats, care regime, biofiltration, filter feeder population, etc. Every week or two is a nice, safe timeframe. A month would be on the far end of the usable range. IMO, it would be preferable to remove the carbon after a month, even if you didn't replace it with fresh carbon right away.
I second (third? fifth?) the idea that running carbon in a phosban reactor is an excellent way to get good use of the carbon. All of the carbon gets good surface contact and there are no channels that develop, like water channelled out the Grand Canyon. Water does the same to packed carbon and eventually you get very little carbon surface area actually touching the water bring filtered. Using a reactor completely eliminates this problem, so you aren't throwing away half your carbon that is barely spent along with the other half which is completely spent.