I'm going to try and give an answer as serious as possible, but I'm no expert.

You separate battlefield and assassinations for the use of poisoned blades, but you have to remember that poisons in general have very different objectives in both cases. In the latter poisons were common, but usually left in food or wine, possibly even coated bed cloths. The objective in this case would be to take out a key figure. Poisons were also very commonplace in warfare, as were other forms of biological warfare, but rather than targeting a single individual, they more than often were a tool of psychological warfare. Greek generals encouraged the projection of snakes by catapults (Hannibal had apparently used a similar technique to cause chaos in Eumenes the IInd's ships) because of how fearful men were of their bite. Far more common was the poisoning of wells and food supplies. This has occurred throughout history, possibly dating back to tribal warfare in our pre-civilizational history. The objective wasn't always to kill, but rather than weaken the opponent, to sap them of their strength for the fight to come. Honor wasn't as widespread as it is in our romantic perception of antiquity and medieval warfare. The reason why I'm not yet taking about weapon poisoning is because poison was a huge part of warfare, whereas blade poisoning is the niche.

The "poisoned weapon" trope is a rather common one in literature, one could almost believe that every major villain and their grandmothers coat their blades in some foreign poison originating in a Latin-american backwater. Yet in reality while poisons were fairly commonplace (or as commonplace as they could be), they had a far more local point of origin. In Europe one of the most common poisons was cyanide, and could be extracted from apricots. But cyanide couldn't be coated onto weapons, it would most likely just cause an irritation of the wound which wouldn't be fatal. As a matter of fact most poisons couldn't be applied to weapons, as a penetration of the blood stream was neither guaranteed (slashes are rather ineffective in actuality) or were simply ineffective in such a use. As such when weapons were coated with poison it was most likely venom. Viper venom was most common in Europe, while in Latin America, Curare was a common plant-based poison that led to a slow death by paralyzing the nervous system. Most areas in the world had access to poisons that could enter the bloodstream that were know to them during the Middle-ages.

This leads to the most important question, which weapon was poisoned ? Honestly, the poisoned dagger shoved in someone's back was rather rare. The most poisoned weapon was always the arrowhead. Arrows (unlike the popular misonception) rarely killed, they just don't pack the punch bullets do, and do little damage beyond simply puncturing holes, possibly in vital organs. They were usually used to incapacitate or stagger. But poisons could finish the job. This wasn't that common in medieval Europe, simply because of the cost (how impossible it would be for a Capetian king to tip every arrow in Viper venom, when he has 20 000 archers, some Genoan crossbowmen, and most of them have quivers with 30~50 arrows or bolts because they can't hit anything in the chaos of warfare) although it did occur to a certain degree in Antiquity. Add to that how slow poisons do kill. Venom can lose it's potency rather quickly, and if your poison takes 30-40 hours to kill someone then it's useless for open warfare (if the arrow can deliver a dose that can cripple someone before he dies, then the arrow would have crippled him in the first case, and the victim will only die once the battle has already ended). In the cases of tribal warfare across the African and American continent though, it was a perfect tool for short skirmishes that mainly left wounded men rather than corpses in it's wake. Scythians used to poison their arrows, but they mainly pushed their reputation as poison-users as a scare-tactic. North-Western Indians also used Viper poison (as far as I can remember) on their arrows during their battles with Alexande, as did Romans in particular cases. But these are more often than not the exception rather than the norm. In truth, unless you were expecting a long attrition war, the tactical advantage given by poisoned arrows on the battlefield was minimal at best.

Now comes the question of poisoned blades in general, since it's the true heart of your question. It simply wasn't used in assassination, if you can't land a strike with a knife that would be fatal to begin with, then it probably wouldn't deliver enough poison to be fatal either (but if you could, then the poison is useless, this is the problem with poisons on weapons). Blades have limited space, and you can't just dip your blade into a sugar frosting of arsenic and allow it to get over all your clothes before you fight, and then scratch someone with it and watch them fall over. The poison loses it's potency rather quickly in open air and allowing it to drip everywhere is more than unsafe. Depending on the origin of the poison, it was usually applied so as to dry on the blade rather than be a liquid ooze that covered it. The Indonesian Kris was actually forged with poison infusions, and was used in semi-open warfare (although today it's mostly traditional, and it didn't have much reach). Swords were often poisoned, as were spears, but this was in primitive warfare, and as war evolved, they were discarded, for more effective weapons that killed instantly. Most poisoned weapon tales are derived from archaeology since by the time written accounts arose, they had already passed out of use.

So was blade poisoning a thing ? Not really, be it for warfare or assassination, it's just impractical, the doses delivered were too light to have any effect unless you just shoved the full extent of the blade into your opponent (if he isn't already dead that is). Arrows were used, but honestly, a pile of dung to fester an infection was both cheaper and as effective. I'm sure there were minute cases, and it was common in primitive forms of warfare, but it would have become obsolete by the time we reached the Medieval Era in the 400s, let alone the Game of Thronesian 13th century you were probably thinking of.

https://www.reddit.com/r/AskHistorians/comments/1rx5o2/did_poisoned_weapons_especially_arrows_ever_got/

This older thread was the closest I could find that might partially answer your question and mostly focuses on whether poison arrows - not blades - were a thing, the consensus seeming to be that they were.