Is AES-GCM suitable for per chunk encryption?
The limit is on a per-key:nonce basis. As long as the key or nonce are changed between messages before the limit, there will not be any problems. Different users are using different TLS sessions (which uses 96-bit nonces), which means different keys and nonces. From Wikipedia's page on GCM:


  GCM has been proven secure in the concrete security model. It is secure when it is used with a block cipher that is indistinguishable from a random permutation; however, security depends on choosing a unique initialization vector for every encryption performed with the same key (see stream cipher attack). For any given key and initialization vector combination, GCM is limited to encrypting 239−256 bits of plain text (64 GiB). NIST Special Publication 800-38D includes guidelines for initialization vector selection.


If TLS is not the protocol in use, then it depends on a few factors such as the size of the nonce (bigger is better) and whether or not it is random or sequential (sequential is better). In general, a competent cryptographer will not select insecure parameters. If you are trying to design your own encryption utility, you should be using an easy to use and foolproof library, such as libsodium.