What you're seeing in that scene isn't a standard tactic, it's an improvised solution to a unique situation.

Generally speaking if you're going to fight a ship, you want as many of your crew as possible on the actual ship doing the actual fighting.

I think that oared boats were used as tugs in harbor; there's a passage in one of Conrad's novels, I think, that describes this.

There's also at least one case of a warship using boats for mobility at sea, although this was to avoid battle instead of during it. Very early in the War of 1812, the USS Constitution mistook a squadron of five British frigates for the rest of the "Original Six" American frigates; when the crew realized their mistake, no wind was blowing and the British were very close. The Constitution launched her boats, which towed her away; the British then launched theirs, which towed them after the Constitution. Eventually, the Constitution dumped 2,000 gallons of drinking water, and with that lightened load escaped after a 57-hour chase.

That said, /u/skiddleybop is correct: this appears to have been a spur-of-the-moment improvisation, and it definitely wasn't something that a ship would do during combat. Remember that during combat, most ships would have marksmen in the rigging, and most warships carried light swivel guns firing grapeshot; a boat's crew would have been extremely vulnerable to those sorts of weapons, while a ship offered more protection.

While at the USS Constitution museum in Boston, I read a display about how that tactic was used when they were being chased or chasing and there was no wind. I recall it was very tiring for the crew but the alternative was to be captured or sunk.