I thought that coinbase was just a cryptocurrency exchange, similar to Binance / Huobi / Kraken. Does coinbase have a special role? Is coinbase in some way bound to follow BIPs or did/does Gavin Andresen have influence on what coinbase does?
The coinbase mentioned in BIP34 is not the company; it is referring to the first transaction in a Bitcoin block (which is special as it is allowed to bring new currency into circulation).
Coinbase-the-company didn't even exist when BIP34 was written (or at least, wasn't known/public). The company was named after this special transaction.