There's Options

Effects

Two items appear after defeating a Boss. Only one can be taken; the other will disappear.

If the boss has a fixed item drop such as The Harbingers, the other item will be selected from the normal Boss item pool.
Two items appear after clearing (Boss Rush) Boss Rush.
Both items that appear after defeating a Boss in (The Void) The Void are from random item pools. Likewise, both items that appear after defeating The Fallen are Devil Room items.
There is no effect on items dropped by Mini-Bosses.
There is no effect on what appears after defeating Mom, even if The Polaroid / The Negative have not unlocked (in which case only 1 Boss item will appear).
Does not retroactively effect (Boss Room) Boss Rooms that had already been cleared prior to picking it up.

(Added in Repentance) Crooked Penny / Diplopia: Both items are copied, allowing Isaac to take both copied items without the other disappearing. The original pedestals will function as normal, removing the other if one is collected.
(Added in Repentance) (Jacob and Esau) Jacob and Esau: Both brothers can take one item each if they are lined up properly.
(Added in Repentance) Satanic Bible: Both items will be Devil Room items; only one can be bought.
(Added in Repentance) Vanishing Twin: The item which appears after defeating the copied boss is also affected.