So, we're training someone to be a fighter.
First, lets consider that mere proficiency isn't the same thing as being a master of said weapons. You simply know how to use them without stabbing yourself in the foot. Well, the game doesn't really model that, but, realistically, if you're not proficient, there's often little reason to use a weapon. Is it possible to train someone in how to use various weapons within, say, 3 months training time (I believe that one season was standard military training time for peasants)? Is that realistic? I believe so, yes. US Marine training is 12 weeks long, and think of everything that they teach you in those weeks.
Secondly, in addition to learning how to swing a weapon, you have to build up muscle mass in order to swing the weapons over the course of a day. You also need to practice accuracy. These are modeled by your Attribute scores in game, and are raised independently of knowing the correct way to swing the weapon without hurting someone. I can have a STR 8 elf wizard proficient in using a longsword. Will he ever actually USE it? Doubtful. There's a bit of meta-gaming considerations here as well - you want a class to grant all these proficiencies so players have the option of making a choice. A low STR, high DEX fighter may never learn more than Finesse weapons and archery from a story prospective, but we need to be able to cover all variations of STR and DEX from a design perspective.
Actual mastery of using a weapon involves learning Fighting Styles and appropriate feats in addition to any class features and a high attribute. Our above elf wizard may be proficient with a longsword, but he'll never have Duelist, pick up a feat for it, get Extra Attacks or any other things you do with weapons. Training often requires raising all those things at once; again, proficiency is an easy part of the instruction.
Third, lets consider that most of these skills are transferable. Instead of looking at long lists of armors and weapons, we really can break it down to more basic things - light, medium, heavy armors, shields, knives, bludgeons, swords, crossbows, bows, axes, thrown. Teaching someone how to properly wear and care for light armor will take hours. Using knives and a polearm are likewise simple. It'll take me a single day to teach proficency in all three of those.
So, no. Its not unrealistic considering how much beyond proficiency is required to use a weapon effectively. If anything, the game puts way too much weight on learning proficiency after character creation.