At the end of Captain America: The First Avenger, Cap wakes up in a "Recovery Room" with an Agent Carter lookalike and all of the good ol' 1945 comforts, complete with a handheld radio playing a baseball game (If I remember correctly, it involved the Dodgers). The only problem was that Cap had already seen that game; it was from 1941, 4 years prior to his crash. The question is, why would SHIELD (Which obviously set up the whole scene) play a game from an earlier time, when they could have played a game from after the crash?
Your question is based upon the premise that the audio track was intended to be circa 1945 and something that he had not heard before. If that was the case, then just about any post-1945 baseball game (with appropriate beer/hotdog/brylcreem commercials) would do.
I would postulate that his surroundings were designed to reduce culture shock and make him feel comfortable if and when he awoke but not intentionally designed to deceive. The audio track playing on a loop through the period-accurate radio was a baseball game they knew he was familiar with and had been chosen for that very reason.
This would parallel reading passages of a favourite book to a coma patient for therapeutic reasons.
SHIELD likely could not predict the exact moment at which Cap would awaken. Thus, they put him in a period-appropriate room, and part of that included the radio.
They probably had the radio playing the whole time he was in the room, not just right before he woke up. Given the amount of time it would take for him to awaken, they likely provided a large number of audio files. Someone who was told to select audio files appropriate to his era must have gotten a bit sloppy with the dates.
Creating a convincing illusion is HARD. They screwed up only in a single, relatively minor way. Yes, they could have avoided it if they had paid more attention, but they had no way to know that was an area they'd have to pay more attention to. They simply didn't have (or use) enough resources to make it a 100% perfect illusion.