A table showing an example is below. There's no cap, with diminishing returns. At 88 Luck the chance drops to 2%.
Luck
Item Pool
Poop/Coal
-10
0%
100%
0
83.3%
16.6%
8
90%
10%
A Lump of Coal and The Poop are fixed drops, and cannot be changed by having Chaos, holding NO!, and other similar effects. This also means Mystery Gift can drop copies of either item even if Isaac has already encountered/possessed them.
Spawning The Poop will create a poison cloud, as if having used The Bean, down to matching the former item's effects.
Synergies
Book of Virtues: Spawns a random wisp. For the rest of the run, a random wisp spawns after clearing a room if the one given by the item was destroyed, including the replacement wisps given by this effect.
Damocles: When Damocles is activated, a second item will be spawned with adherence to the item pool for the current room. Even if Mystery Gift "fails" and spawns either The Poop or A Lump of Coal, a second item from the current room's item pool will always be rewarded.
Interactions
Car Battery: No effect, Mystery Gift will only spawn a single item.
Void: Like all single-use active items, this item cannot be absorbed by Void. Instead, it will consume and use the Mystery Gift.
Mystery Gift will be absorbed and will spawn the item upon Void's next use.
The appearance and pickup quote of the item is a reference to the game Spelunky, where there is an item called "Mystery Box", with the description "Thanks! I wrapped it up nice for you!"
The item's name is also a reference to the Pokémon franchise's feature for distributing event Pokémon.
The item having a chance to give A Lump of Coal references Santa's tendency to gift coal to naughty children.
The item having a chance to give The Poop references one of Isaac's floor transition's dreams, in which he is gifted poop.
A Lump of Coal or 
The Poop instead.
Chaos, holding 
NO!, and other similar effects. This also means Mystery Gift can drop copies of either item even if Isaac has already encountered/possessed them.
The Bean, down to matching the former item's effects.
Book of Virtues: Spawns a random wisp. For the rest of the run, a random wisp spawns after clearing a room if the one given by the item was destroyed, including the replacement wisps given by this effect.
Damocles: When Damocles is activated, a second item will be spawned with adherence to the item pool for the current room. Even if Mystery Gift "fails" and spawns either 
Car Battery: No effect, Mystery Gift will only spawn a single item.
Void: Like all single-use active items, this item cannot be absorbed by Void. Instead, it will consume and use the Mystery Gift.
Mystery Gift's appearance prior to Booster Pack #2.
