Bloody shoulder sounds awful, but it's just an unusual pattern that forms on some grey horses.
The horses affected are what are called "fleabitten" grays - they keep tufts of their base color for a while as they lighten. Sometimes these tufts join together to form larger patches - most often on the horse's shoulder, hence the name. When found elsewhere they are called blood marks. They often look red or red-brown.
These markings are more common on Arabians simply because the fleabitten grey pattern is particularly common in the breed. The Bedouin consider blood marks a sign of quality, partly because they are pretty much impossible to breed for - you can breed for the flea bites, but blood marks are pretty rare.
Bloody shoulder markings can actually get bigger with age. Also, sometimes horses will go dappled and then develop flea bites, which can result in this:
The brown color of the blood mark, which is extensive, and the dark points tell me this horse is bay "under" the grey. Image source, Kumana via Wikimedia Commons.