Why AC systems grow mould in this climate
An evaporator coil works by being colder than the air passing over it. Humid air hits cold fins, water condenses out, and that water runs down into a drain pan and out through a condensate line. The coil is therefore wet whenever the system is running, by design. Add the fine dust that gets past a filter, and you have a permanently damp surface with a food source on it, sitting in the dark. That is close to ideal for mould and bacteria.
In the UAE this runs for most of the year rather than for one season, and the humidity load is high, so there is more condensate and more hours of wet operation than the same equipment would see in a dry climate. Two failure modes then make it worse. A condensate drain that is partially blocked leaves standing water in the pan instead of draining it, which turns a damp surface into a reservoir. And a system that is oversized for the space cools the air to temperature quickly and shuts off before it has removed much moisture, so it short-cycles: the coil never dries out between runs and the room stays humid.
Filters matter more than people expect, because the filter is what keeps the coil clean. A clogged or bypassed filter lets dust straight onto wet fins, where it packs into the coil and becomes the nutrient layer that growth establishes in. Once a coil is fouled it also restricts airflow, which lowers coil temperature further and increases condensation, the problem compounds.
The result is the smell people describe as musty, damp or "like a wet cloth", strongest in the first minutes after start-up because that is when the accumulated air in the system and the disturbed surface release the most. Where a system serves several rooms through ductwork, the same growth can be distributed to every outlet it feeds.

