As per my understanding, a Universal Hash Function isn't a cryptographic hash function & it's output isn't uniformly distributed. However, this is still secure because it's actually a family of functions & one or more of the random inputs to the function decides which function is actually picked from the family of functions & this is what makes it secure.
However, these are the Parameters to GHASH
$GHASH(H, A, C)$ where
$H = E(K, 0^w)$
$K$ is the encryption key & it's fixed, so a new one isn't picked every time, which means $H$ is also fixed.
$A$ is the Additional Authenticated Data
$C$ is the Ciphertext
So how exactly is this a universal hashing function - what is the family & how exactly are we randomly choosing from the family?