A worn-out old duster. The coat protects your clothing from trail dust, but not much else.
It is a worn, dark brown, oil skin duster. This type of coat was usually worn by deputy lawmen of the time, likely the reason for its requirements, seen below. It was also a popular choice for bounty hunters.
Dusters have a long history in the West and Western movies, so there is likely no single inspiration for this outfit.
A true duster has a long slit at the back, almost up to the waist, dividing the tail into two long flaps so that it can be worn easily when riding. The duster appearing in Red Dead Redemption has no slit and would in reality be very impractical for riding a horse. This design may have been chosen to simplify character animation.
In Redemption II however, the player can buy dusters that have this trait fixed.
This is true if the coat is worn buttoned-up. Unbuttoned, as Marston wears it, it acts similar to a cape, which can be worn horseback without the need for a slit.
Even when playing as Jack Marston, John's Cowboy attire is still worn underneath the duster coat.