Is the length limit of AES-GCM per key or per (key, nonce) pair?
It is per message. I.e. per (key, nonce) -pair.

The reason it exists is that GCM uses CTR mode for encryption with (normally) a 32-bit counter. That means you can only encrypt $2^{32}$ blocks, i.e. $2^{39}$ bits with AES. The 256 bits that are subtracted are due to authentication.

So you can encrypt multiple maximum-length messages securely.