You can add a player specific Birthright description using the following function:

EID:addBirthright(characterId, description, playerName, language)

The arguments playerName and language are optional.

Example usage

local myPlayerID = Isaac.GetPlayerTypeByName("Funny Test character")
EID:addBirthright(myPlayerID, "This is a custom birthright description for my character")

If the player should get his own icon for his birthright, you need to create a new icon that is named "Player" following by your PlayerID:

local myPlayerID = Isaac.GetPlayerTypeByName("Funny Test character")
local icons = Sprite()
icons:Load("gfx/myIcons.anm2", true)
EID:addIcon("Player"..myPlayerID, "MyAnimation", 0, 16, 16, 0, 0, icons)