Well, your answer likely depends on what era, region, and kind of fortification you’re asking about. Much of what I will say I’ve gathered incidentally into memory from many sources on Medieval European warfare, but as a general overview and source for some specific examples, here is a paper, published in the Journal of Landscape Ecology, which deals with waste in general for rural/urban populations as well as fortifications in Europe in some detail and provides other sources for you to follow up with.

The castles of the High and Late Middle Ages generally relied on unimaginative solutions like latrines and literal cesspits, which are effectively just pits for depositing waste. Sometimes lye would be added in a layer on top to mask the odor but not always. Nicer castles however often had one or several rooms called Garderobes, which were effectively a lavatory that just... well, it was an elevated toilet seat above what was just a sluice. It would be located adjacent to an outer wall so that right outside, underneath the sluice, a cesspit could be dug, although sometimes instead of a cesspit there was a channel that would lead to the moat or in some instances, the garderobe would be extended over the moat so that waste was directly deposited there.

“Wait, but what did they use to clean themselves afterward? What about toilet paper?” I hear you pondering but not asking out loud - the Romans had the xylospongium/tersorium, what of later societies? Well, at least one paper has suggested based on the remnants found within those cesspits below garderobes that moss was used as a toilet paper by commoners who couldn’t afford cloth like their elite peers, and moss may have found more general use during a siege when new cloth couldn’t be acquired.

Now, not all waste was disposed of in this manner, particularly in towns. Much was instead collected in cesspits (or simply buckets) and used as fertilizer in gardens in whats referred to as “night soil”. While very ecological and more purposeful than a cesspit, and not requiring any kind of indoor plumbing, this did have the unfortunate side effect of breeding immense amounts of bacteria and disease, especially of enteric parasites like helminths and worms, as the handlers who collected night soil would be exposed to the dormant parasite eggs in the waste, accidentally consume it when it ended up on their hands or distribute it to others on their hands wherever they went, keeping communities sick for very long. In fact, there was a major push in American public health at the turn of the last century around limiting night soil use in the southern states to prevent hookworm, with the argument at the time being that hookworm was both so endemic and its morbidity so high that it was causing terrible loss in economic productivity to those states, but I digress immensely.

What of the various animals that were kept within the castle’s walls? Well, some of that waste could be used as fertilizer for the castle gardens, however the gardens were generally very small and usually only grew herbs to aid the kitchen chefs and weren’t useful at any scale or caloric amount for provisioning a garrison - for that defenders were effectively tied to whatever was in the storehouse and eventually when that ran out whatever animals were in the castle. Instead, waste, varying from human and animal excrement to animal carcasses, would sometimes be loaded into barrels and used as artillery stock to bombard besieging enemies. This apparently happened during the siege of Karlstejn Castle in 1422.