He's a chocobo, not a monster. He's called Bobby Corwen.
A chocobo egg is found by two black mages, Black Mage No. 33 and Black Mage No. 111, who take it to their home in the Black Mage Village and watch over it. They stop Quina as s/he tries to eat the egg when the party visits the village. The two black mages do not join the others with Kuja, because they are busy looking after the egg.
The egg hatches while the village is empty, all except Zidane's party, the two black mages and Black Mage No. 288. The black mages raise the chocobo chick with love and affection and name it. Upon its hatching, the two black mages experience an overwhelming happiness. Vivi also feels this, and thus believes he and his fellow black mages cannot be puppets since they can experience such joy.
The mages refer to the chocobo by his full name, much to the annoyance of Eiko. One of the Genomes believes Bobby to be a monster as he is unaccustomed to life on Gaia, but later grows attached to the chocobo.
Bobby Corwen appears with a Wind-elemental card.