They mean the same thing but are used according to the gender of the noun they are possessing. Elefant is masculine, so you would say "elefanten min". Jente is feminine, so you would say "jenta mi". And dyr is neuter, so you would say "dyr mitt". It's just the same as when you have to use en, ei, and et, but they all mean "a".
It's either "Det er mitt dyr." (emphasize posessiveness) or "Det er dyret mitt" (normal possessiveness). So the noun is definite if put in front of the possessive.
You can think of this as "It is my animal" vs "It is the animal (of mine)" (while this is unnatural in English, it is the natural form in Norwegian).