For instance, in the Palace of Holyroodhouse, their display on Mary, Queen of Scots, had some pieces of artwork (commissioned by Lord Darnley, I think; they didn't allow photography, so I may be misremembering some of the details). The display said that the artwork was laden with symbolism that would have been immediately apparent to anyone of the time.
It even claimed that the meaning behind the depiction would not have been particularly debatable or open to misinterpretation at the time, seeming to imply that the imagery could be read almost as effectively as we might read a paragraph today.
Was there this sort of universal visual language that was employed back then, in a way that has fallen into disuse in the face of ubiquitous literacy?
I think saying it was a universal visual language might be going too far, but visual cues were certainly employed in artwork to convey the story or meaning to non-literate audiences.
The example I'm most familiar with is the Bayeux Tapestry, the 230-foot long embroidered cloth, created in about the 1080s to commemorate William's conquest of England in 1066. The Bayeux Tapestry actually has captions woven into it, but they're entirely in Latin (so the average person seeing it hanging in the church probably wouldn't be able to read it). The tapestry tells the story of the conquest, starting with Harold Godwinson's trip to Normandy. Harold was a noble in England under the reign of Edward the Confessor, and had his eye on the English throne. Harold is taken prisoner by Guy down in Normandy, but William (Duke of Normandy at the time) gets him released and they band together to attack the castle of Conan, a nearby duke. During that sequence, William does something which we today don't really blink at, but which would be extremely important to a medieval viewer: he gives Harold arms and armor. Vassals were often given arms and armor by their lords, so a medieval viewer looking at this would get the subtext of "Huh, Harold is basically his vassal now". The Bayeux Tapestry was created to celebrate Williams' triumph over Harold at Hastings, so it makes sense that the author(s) of the tapestry would want to legitimize the conquest by effectively saying "Harold was his vassal all along, so Harold seizing the throne of England was an act of rebellion."
This is reinforced a little later in the text when Harold swears an oath of fealty to William on holy relics, making the implied vassal/lord relationship more explicit. Even we still understand the significance of swearing on relics.
There's other smaller hints in the tapestry too that help tell the story: the Anglo-Saxons (Harold and his men) tend to have mustaches while the Normans (William and his men) tend to be clean shaven, though it's not a constant.
It's also tempting to read significance into scenes that we just don't understand. At one point a cleric appears to be striking or touching a woman named Aelfgyva, and I haven't heard or read of a satisfactory explanation for what that scene means and why it was significant enough to narrate. There's also the little naked man down below; is that related to the scene above? Would a medieval viewer in Bayeux in the 11th get something extra out of this that has been lost to time?
So, really long-winded way of saying: Yes, I think there was definitely a visual language in medieval art, though it was probably not completely universal. Meanings could change from region to region, but I think there was a strong tradition of symbolism which could be "read" by illiterate members of society.
