All pickups on the ground become their doubled variants as long as Isaac's held pickups are all the same number, e.g. 65 pennies, 65 bombs, 65 keys.
Notes
All single keys, bombs and red hearts in a shop will become free when doubled.
This is likely due to the entity being removed and a double variant being spawned, instead of simply overriding it before it spawns like Humbleing Bundle does.
Upon picking up a non-doubled pickup that makes Isaac's pickups equal in number, a doubled variant will be spawned as well.
Doubled pickups will not revert if Isaac's pickups become unequal.
This trinket works well in Greed mode, doubling all pennies spawned from completing waves as long as Isaac's pickups are equal.
Strategy
Previously seen non-doubled pickups WILL become their double variants after reaching equal numbers for all of Isaac's pickups. Said doubled pickups will never revert back even if Isaac's pickups numbers changes or if the trinket is dropped.
However, changing any number will prevent doubled variants from spawning until equality is reached again.
Said this, it is highly recommended for letting doubled pickups on the ground until going to the next floor.
Heart pickups will also be doubled, however Isaac's hearts amount will not prevent Equality! from working. This means doubled hearts can be picked up as much as you want without any downside.
IV Bag, especially if paired with Old Bandage, can be abused for doubled pennies to spawn while also taking advantage of said doubled hearts.
Options?: While Equality! is active, if on of the two rewards are doubled pickups, both rewards can be collected without the other despawning. Options? also makes Equality! easier to keep active.
Tainted ???: T.??? will always have 0 bombs, making the trinket only possible to activate while also having 0 coins and keys (except with the Bombs are Key pill).
Tainted Keeper: Doubled pennies from killed enemies will not despawn. All other coin types will still disappear.
All single keys, bombs and red hearts in a shop will become free when doubled.
Humbleing Bundle does.
IV Bag, especially if paired with 
Old Bandage, can be abused for doubled pennies to spawn while also taking advantage of said doubled hearts.
Hypercoagulation / 
Tainted Magdalene: As long as Equality! is active, all hearts spawned their respective sources will be 
Double Hearts and do not disappear.
Options?: While Equality! is active, if on of the two rewards are doubled pickups, both rewards can be collected without the other despawning. Options? also makes Equality! easier to keep active.
Tainted ???: T.??? will always have 0 bombs, making the trinket only possible to activate while also having 0 coins and keys (except with the Bombs are Key pill).
Tainted Keeper: Doubled pennies from killed enemies will not despawn. All other coin types will still disappear.
The Binding of Isaac: Rebirth
