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Here is the list of characters that currently have their extra Birthright functionalities and their descriptions:
Isaac: D6 becomes a pocket active item if you have one. Rerolling items has a chance for item from Treasure, Devil, Angel or current room pools.
Magdalene:
Chance to charm nearby enemies. Yum Heart heals 2 red hearts instead of one.
Speed is linked to number of red hearts. Gains 1 soul heart on new floor.
Cain:
Enemies have a chance to drop random pickup (configurable for unlockable pickups). Champions drop two of the same type. Chance to spawn an item from room's itempool if there was a boss.
Golden key on every floor.
Judas:
Spawns 3 nickels. Chance to gain +5.0 damge and Brimstone effect for current room.
Using books activates Necronomicon effect. (Doesn't work with Book of Virtues and How to Jump. Modded active items that have tag 'book' can be added. See 'Readme.txt' in mod's folder.)
Dark Judas:
When getting hit, damages all enemies in room similar to effects of black heart, Necronomicon and The Negative.
Same as Judas' Antibirth Birthright.
??? (Blue Baby): Number of soul hearts give stat ups. Non soul (black) hearts become soul hearts or other pickups (cards, runes, keys, bombs and others). Compatable with Double Soul Hearts mod.
Eve: Getting hit will occasionally spawn additional Dead Birds that persist between rooms
Samson: +4 damage up. Gets ability to pickup rocks (and other objects) by pressing "Drop" button (Left CTRL on keyboard or RT on controller by default) and throw them. (Yes, it's Mom's Bracelet without Mom's Bracelet.)
Azazel: 20% chance to shoot in 8 directions on hit.
Lazarus: Coming soon.
Eden: 25% chance get random passive item effect when entering room.
The Lost: The Lost: revives with copies of all of his passive items doubled (except birthright and quest items), but he can't have holy mantle effect. You can get mantle effect by using Holy Card but only for current room. After that effect disappears..
Lilith: Spawns 2 mini Isaacs when entering hostile room, 4 - big room (L-shaped included). Box of Friends now affects mini Isaacs.
Keeper: One free item that costs money per floor, effects of Silver Dollar and Rib of Greed.
Apollyon: Void becomes pocket active. Voiding passive items with quality higher then 0 will give random extra stat ups. Higher quality items give more stats.
Bethany: Curses become blessings and always appear on floor, even if floor wouldn't have one.
Blessing of the Prevailing: 20% chance for red doors to appear. Oposite of Curse of the Maze.
Blessing of Enlightenment: Bethany has a large aura around her that burns enemies inside it. Oposite of Curse of Darkness
Blessing of Order: 30% chance to teleport into unvisited special room in order: Treasure or Planetarium, Shop, Secret or Super Secret, Devil or Angel, Boss. Oposite of Curse of the Labyrinth.
Blessing of the Baptized: 35% chance to transform into a random character when entering a room. Can't transform in Greed mode and when entering Boss, Boss rush, Black market, Isaac's room, Error, Angel and Devil. Can't transform into ???, Dark Judas, Keeper, The Forgotter, Tainted Isaac, Tainted Judas, Tainted ???, Tainted Lazarus, Tainted Keeper, Tainted Forgotten, Tainted Bethany. Oposite of Curse of the Unknown.
Blessing of Foresight: after 5 seconds activates Glowing Hour Glass effect when entering for the first time into hostile, Treasure, Shop, Curse, Devil rooms. Reviels hidden item in alt floors. Oposite of Curse of the Blind.
Blessing of Clarity: full mapping effect. Oposite of Curse of the Lost.
Blessing of the Virtuous: 5% chance for tears morph enemy into random wisp. Oposite of Curse of the Cursed.
Blessing of the Imperceptible: doors are always open and effect of Camo Undies. Oposite of Curse of the Giant.
Jacob: Esau gives half of his non story passive items to Jacob in exchange for Red Stew
Esau: 100% chance for extra reward spawning after clearing room. Devil rooms have flat 50% chance to spawn no mater what items you have and no matter if you had red heart damage. Angel rooms can't appear.
Tainted Isaac: Spindown dice becomes his pocket active (like in Four Souls) and using it has a chance to activate Soul of Isaac effect.
Tainted Magdalene: Only combined variant. Every cleared room restores one red heart.
Tainted Cain: He pick up 1 item on current floor, he have a chance to pick up Birthright without reducing it into pickups, if Tainted Cain doesn't have one. Can be modified in Mod Config Menu.
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Tainted Judas: Changes Dark Arts to Twisted Arts: an active item that has longer duration than Dark Arts and freezes time in a room for few seconds.
Tainted ???(Blue Baby): Chance to poop out friendly dip. Having special items can give different type of dips.
Tainted Eve: Has a chance to spawn additional clot of the same type. All clots take more hits to kill.
Tainted Samson: Damage up. Damage depends on how many enemies are around Tainted Samson in radius of 100 and how many times Tainted Samson has been hit after picking up Birthright. Last for floor.
Tainted Azazel: Gets Brimstone effect and flight.
Tainted Lazarus: Flip charges over time.
Tainted Eden: Coming soon.
Tainted Lost: Coming soon.
Tainted Lilith: Spawns red creep that deals 250% player's damage. Every time umbilical baby(fetus) is shot spawns red creep trail that deals 150% player's damage.
Tainted Keeper: Coming soon.
Tainted Apollyon:
Spawns 20 random harbinger locusts on pickup. Entering hostile room will spawn 2-5 random locusts.
Abyss locusts have small Godhead aura.
Tainted Forgotten: Coming soon.
Tainted Bethany: Only combined variant. Item wisp shoot homing tears, that have her tear range and 40% damage.
Tainted Jacob:
Birthright gives +1 Health up, +0.3 Speed up, +50% Damage Multiplier, +1 Tears up. Damocles effect for Treasure, Secret/Super Secret and Devil rooms when entering them. Dark Esau chases enemy that damaged you before and kills Tainted Jacob in one hit no matter the form (normal or ghost). Having Isaac's Heart will make you a ghost even if you have Holy Mantle (and prevent from one hit KO).
Dark Esau becomes friendly. Targets enemies instead.

You can toggle them with Mod Config Menu or command prompt.

Ah, yes. Console commands:
extrabirthright [command] [args]:
[command] - help, change, get or default. Use full command "extrabirthright help" without quotes for info in console.
[args] - [character] [option]:
[character] - after help command type charlist for list of characters, for other commands - name of character from charlist (WARNING: CASE SENSETIVE). Example: Isaac, Magdalene, Lilith. For tainted charcters add "Tainted" before name.
[option] - option's number from 1 to 3 (Original Effect, Mod Effect 1, Combined Effect 1). For Magdalene, Cain, Judas, Dark Judas and Tainted Jacob from 1 to 5 (Original Effect, Mod Effect 1, Mod Effect 2, Combined Effect 1, Combined Effect 2).