Knights & Kaiju is at its core a Pendragon campaign. Knights carry swords, usually. Knights in a K&K campaign aren't much different. Nobility carrying melee weapons as both defense and fashion is a standard trope.
But what about wearing a proper knightly sword while crammed in a cramped and sweltering cockpit? Not likely to be practical, even if we want to disregard comfort for fashion. What would a proper knight of our post-apocalyptic 80s world wear for personal defense and honor? Due to the cramped nature of the mecha cockpits the personal arms of knights and other men-at-arms, such as squires, consist not of swords, laser or otherwise, but of weapons that not as awkward within the confines of a cockpit. These are by tradition daggers and knives, used mostly for those few times a knight fights another knight sans mecha, and lacking the technical base to manufacture personal scale energy weapons, light slug-thrower pistols, which are used to deal with unruly peasants like bandits.
The traditional courtly attire of knights are also influenced by the cramped and hot cockpits. Clothes that would be associated with medieval knights, rightly or wrongly, like long pointy shoes, huge sleeves, poofy pants, and neck ruffs are right out because they would get in the way; likewise, tight and constraining clothes are just as bad because of their tendency to hold too much heat close to the body and cutting circulation off to extremities after long periods of operation. (In case it wasn't apparent by now, I am taking a lot of cues from Battletech regarding how Mechwarriors dressed while piloting, which is mirrored slightly in how the members of Tekkaden from Iron-Blooded Orphans dress while piloting Mobile-Suits/Workers while on a planet.) By their nature, those mecha used for knightly pursuits have enclosed cockpits and produce a lot of heat while in operation, forcing knights to wear as little as possible. In short, members of the aristocracy and their warrior retainers dress like post-apocalyptic warriors. Picture Conan or Thundarr, only without the fur, or Sean Connery in Zardoz, or the Spartans in 300. That's right, knights universally walk around in speedos and bootyshorts with a light harness for carrying minor items, and a cloak or cape when not piloting. This is a post-apocalyptic world from the late 80s and early 90s, and everybody dresses like it.
In contrast with knights, peasants, not having to pilot stifling hot warmachines, dress pretty much how you would expect peasants to dress.