What you want to do (splice a flight within Europe in the middle of your trip) is no problem - any more than it would be for a French visitor to New York to book a round trip from JFK to Chicago in the middle of their New York trip.
You will be using your passport as your ID at the airport, and probably will have to input passport information when you book your ticket. Paris and Prague are both in the Schengen zone, so there will be no routine immigration checks (passport control). And they are both in the EU, so there is no customs.
The problem comes when people try to book flights on separate tickets on the same day - for instance, US to Paris on Norwegian, then Paris to Prague on Air France just a few hours after arrival. If there's any disruption and they miss the second flight, they are on the hook financially (no refunds - they have to buy a new, last-minute priced ticket) and logistically (they have to figure out themselves how to get to Prague, with no help from Norwegian or Air France). As long as you're not doing this, you'll be fine.