Why is it not "Saya punya kucing banyak"?
Why does banyak have to be before kucing here?
"Banyak" is always put before the noun, just like the word "many" or "lots of" in English.
"saya punya banyak kucing-kucing" should also be marked as correct
No, this is incorrect. When you use a numeral, such as "banyak" or a number, you can't reduplicate the noun.