I have a theory that the reason for this is fairly simple: miscommunication.

"Medieval" usually refers to the European Middle Ages, a period that stretched from the Fall of Rome in 476 until the Fall of Constantinople in 1492 1453. There are about a thousand differing start and end dates, but using the demise of two Roman empires is convenient and a little poetic -- because, unlike dating it to Gutenberg's printing press, we don't have to quibble over whether to point at when he started or the vague "c. 1525" figure that better represents the spread of the printing press. But what's important here is that, no matter the exact start and end dates, the period corresponds to a stupidly long period of time. If we go with a more conservative figure of dating the start to 800 when Charlemagne founded the Holy Roman Empire, we're still looking at a period of nearly 700 years from start to end. More time separated a Genoese soldier on the walls of Constantinople in 1492 from Charlemagne than what separates this sub from Columbus himself. That's the first thing that makes communicating on this subject rather difficult at times.

There's also the matter of nomenclature. How do you define "peasant," exactly? For most people, especially those who don't study medieval history, a peasant is probably just someone who occupies the second lowest strata of society (being just one step above slaves). A farmer, basically. Maybe a fisherman or hunter.

If we look at the English at Agincourt, 5/6ths of their army could probably be classified as "peasantry." Yes, the longbowmen absolutely did go to war with a fairly extensive kit and a very specialized skillset. Yes, those longbowmen had to meet various requirements to actually qualify for service (like actually being able to draw and fire one of those massive warbows). Yes, those longbowmen would have been paid for their time and entitled to various rights while on campaign (most notably looting). But they weren't men at arms (who could very well have held sizable tracts of land) and they absolutely were not of the proper nobility (who comprised, perhaps, 100 of the 6,000 English soldiers at Agincourt).

Now let's return to the first sentence in this post. Even if we get past all the traps above, the miscommunication still arises from the fact that when I say the English army was largely comprised of the peasantry, I'm referring just to social standing. If someone who is not as well informed on the subject were to then read that part of my post and see me classify the English army as peasantry, said person could, not knowing any better, simply conjure up the stereotypical image that Hollywood likes to give us of the peasant: dirty, dressed in earthy tones (bonus points for dirty brown), and probably featured on a set that is just as drab-looking as they are.

There are probably another half dozen factors that would contribute to the miscommunications that seem to breathe new life into these misconceptions on a regular basis, but I think that you've got the idea.