Its Hawser laid rope alright but it does not look at all old and you can still buy it easily.
Its almost certainly some kind of artificial fibre. Easily confirmed by seeing if it starts to melt with a lighter/match in some discrete part.
There isn't anything you can do to keep mould off rope kept in the outdoors. I doubt it makes any damage/weakness to the rope. Don't forget hundreds of outdoor centres have ropes courses where similar rope is used outside all year round.
The mould is likely to be feeding off detritus caught within the strands. I can;'t see mould - a fungus - eating into the fibres if they are artificial.