So, this crossed my mind while watching the recent Game of Thrones episodes. (Note: this post does not contain spoilers.)
How familiar were knights supposed to be with coats of arms and names of noble families? Specifically, I'm asking in the context of wars and battles. Was a knight supposed to know all the coats of arms and banners of all the other knights in his army, so that he didn't strike down an ally by accident? Or maybe there were some noted cases of 'friendly sword stabs' and 'friendly axe chops' (sorry) during battles because someone didn't know that 'Two Longaxes Beneath a Black Crown' fought on the same side?
Also, would a noble family's child be expected to learn by rote memorization all the names and coats of arms of families they might come across on a regular basis?
Yes, a noble was supposed to know the coat of arms of at least the important lords, at least in his own army (because it's rather easy to deduce who's an enemy when you know who's your friend).
The most notable example for this (or rather for it's absence, from which a presumed knowledge for nobles can be inferred) is the Battle of Worringen in 1288, the battle with the largest amount of knights in the middle ages.
The leader of the one side was Siegfried von Westerburg, Archbishop Elector of Cologne, who sported more armored riders than soldiers on foot, making infantry for him rather negligible mostly used to guard the baggage train. On his opponents side fought the people of Cologne, who wanted to weaken the position of the Archbishop to better their own position within the city politics (the Archbishop had a very strained relationship to the citizens of Cologne, with one of his predecessors having been evicted from the city by force at one point).
Most of these people of Cologne were fighting on foot and were not part of the patricians but regular citizens. Now, this militia of citizens had no formal training and very little "noble" education - and subsequently, when they entered the fray, they attacked everyone (or rather: all armored riders) indiscriminately, which is attributed to them not knowing friend from foe because they could not recognize the colors of the nobles.
Another factor for why knowing the colors of ones enemy was important was that battles in the Middle Ages were not really fought to the death. The goal was to take as many of the enemy knights as prisoners to ransom them to pay for the massive costs of waging a war. Knowing who is who is rather essential if you want to know who needs to be spared at almost all costs because a high ransom is to be expected.
So to summarize: Yes, a noble was supposed to know the colors of his friends at least and of his enemies to a large extent when he went to war.
Sadly I can't give you a source for it in English, but on the off chance that you can read German, this information is taken from:
Lehnart, Ulrich: "Die Schlacht von Worringen 1288. Kriegführung im Mittelalter. Der Limburger Erbfolgekrieg unter besonderer Berücksichtigung der Schlacht von Worringen", Frankfurt 1993