In GCM, GMAC, the additional associated data (AAD) (as well as outputs of subsequent blocks) is multiplied by a polynomial H in GF(2^128).
Actually, it's the other way around; the AAD and the ciphertext is converted into a polynomial, and that polynomial is evaluated at a secret point H (in $GF(2^{128})$
This appears to be doing some secure hashing.
Nope, no secure hashing here; if someone knows what $H$ is, they can easily find collisions, preimages, etc. Hence, a full specification of the algorithm (including $H$) is not secure.
This is, instead, a similar sounding object known as an 'almost universal hash'; what that means is that, for any two distinct messages, they get mapped to the same point for only a tiny fraction of the keys (which is the $H$ value here) [1]
That's the sole property that an almost universal hash needs to satisfy; note that it may be easy (for example) to deduce the value $H$ by submitting a message and examining the hash value. Hence, by itself, it doesn't meet the criteria of a secure hash.
What are almost universal hashes good for? Well, they can be used as a component in a Message Authentication Code - in essence, the almost universal hash processes the message (based on a secret key), and then some object with real cryptographical properties (in the case of GCM/GMAC, AES) disguises that hash. This is known as the Carter Wegman construction, and is in common use (Poly1305 would be a second example).
And, one good thing about almost universal hashes is that we have constructions (of which GH is one) which provably meets the criteria.
[1]: Actually, for how GH is used within GCM/GMAC, you need a slightly stronger property; we'll skip this for now.
