The Animal Kingdom counts years from the day of the Ark Fantastic marooning on land. This is an event that signalled the end of a great flood and of course holds a big significance in their society and mythology.
The years before the flood are lost to everyone and unmentioned even in the oldest tales, but in art and literature it is usually portrayed and referred to as an idyllic heaven-like state where animals lived in harmony as there was no conflict and everyone knew their place. The game begins in 1883AA (After Ark), which is 1883 years after the flood subsided and the animals founded the Animal Kingdom they live in today.
You can read more about the day of the New Year in the Animal Kingdom in our blogpost about the Day of the Bells here!