The strata manager is probably incorrect.
In almost all schemes – you will of course need to check your bylaws – the balcony area is usually common property, even though it is intended for exclusive use of the lot owner.
Therefore, this issue of water penetration from one piece of common property to the other is an OC problem: the EC can either elect to remove the mould on an ad-hoc basis or address the drainage issue more permanently.
The latter is usually preferable – water pooling on concrete permanently tends to damage the reinforcing over time and lead to very expensive concrete cancer which can spread throughout the building. Obviously if the balcony is timber, then of course it means the timber is prone to rot.
I would talk with the EC if possible to avoid the Strata manager altogether.
Good luck