What are the strongest arguments for and against treating Coinbase as a true exchange?

If Coinbase were an exchange in the classical sense, it would have a market, where Bids & Offers would be made. You'd be able to buy/sell Bitcoins against various currencies as per your liking and depending on the liquidity in the market, the matching engine would then try to match (pair) your transactions.

Even Coinbase doesn't cite themselves as an exchange: