Individual Firm and Market Demand Curve
What you're doing when you calculate $dQ/dP$ is assuming that all forms change their process by $dP$. This is the same as saying that the market price changes by $dP$ and is why you get the same number.

The main difference here is that when the book says price elasticity of demand is higher for an individual firm, it means that, holding the prices of other firms constant, a 1 percent change in a firm's price would result in a greater change in quantity demanded than when all forms change the price.