It's like how "bike" is short for "bicycle," or "phone" is short for "telephone." "Zoo" is just a more casual and convenient way to refer to a place where animals are kept and displayed for public viewing.
What is the full version of zoo?
There isn't a "full version" of the word "zoo" in the sense of a longer, more complete word. "Zoo" is already a shortened form of the word "zoological garden".