I beat Satan with The Lost but didn't get the item, only the achievement. Why did this happen? I am wondering if I have to actually win the run (beat The Lamb) in order to get the item. Did something bug out or is it part of the game. Also, it was not a seeded run. Thanks!
If what you're saying is that you received the achievement "The Mind", but the corresponding item didn't appear in the item chart or in game, that's how it's supposed to be.
Beating a boss only unlocks the item for the future, meaning that after beating the boss, you can find the item in subsequent runs.