What is happening is that, however much you rinse and squeeze/wring the mop out, traces of whatever (often/usually including our shed skin, ubiquitous and entirely natural) was on the surface(s) you used the mop on remain in the mop and then bacteria and fungi proceed to digest those materials so long as there is any moisture in the mop. Unlike in warmer countries or where dwellings are properly heated so that the atmosphere is drier than here, in Britain things like mops, cloths, etc. rarely dry out completely so the process is ever continuing. This is exactly the same as happens in cleaning cloths (e.g. kitchen), towels, clothing etc. and it is relatively common for kitchen cloths to then leave a very active film on worktops as they are wiped with a cloth that is overdue for a wash (can be as little as 24 hours, all depending on conditions). That film will then smell until the surface dries completely.
The answer is a thorough wash using effective detergents, preferably in warm water, followed by thorough rinsing. It is not necessary to discard the object in question unless effective washing is not achievable. Your mop is probably not the easiest thing to clean but standing it in a bucket of hot water and detergent and then rinsing well with hot water may well do the trick.