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No. It wasn't common to carry multiple flintlock pistols.
A flintlock pistol is heavy, single shot, highly inaccurate and expensive.
Heavy: this varied of course, but you can assume anything from 1 to 3 kg. If you carry two or more pistols, you'd be carrying a lot of useless weight. Blackbeard carried so many weapons not for self-defence, but simply to impress his victims.
A Colt M1911 weighs 1100 grams. An average flintlock pistol from the 1800s weights about the same. That's two centuries of modifications and improvements later, or 400 years in case of the 1911.
Highly inaccurate: what do you miss on almost all flintlock weapons? Sights. A musket is already inaccurate as it lacks rifling, but it can (very limited) make up for it by having a longer barrel. A pistol lacks even that. No rifling, a round ball that comes out of a very short barrel, compared with a musket. You'd be lucky to hit a barn. Even if you stand inside one.
Expensive: Before the industrial revolution, firearms were handmade and consequently expensive. Guns were more commonly produced and less expensive. Pistols were far less in demand, and therefore more expensive.
@Pieter Geerkens is correct, some cavalry units used pistols as their primary weapon. However, they carried those heavy weapons in holsters on the saddle. Rarely on their person.
Those pistol armed cavalry units weren't that great. On the Battle of the Mookerheyde a German unit armed with pistols was defeated with ease by Spanish lancers when those Germans were reloading. (Later in history, exactly the reverse happened.)
In most cases, it was far more practical to carry something else than a pistol, let alone multiple pistols.