Yes, the video is of a French Navy SINKEX in 2004. The target was EE Vauquelin, a 4,000t decommissioned destroyer built in 1956.
According to this website, it was hit by:
– 2x 500lb Paveway LGBs from Super Etendards
– 1x MM38 Exocet from Duquesne
– 3x SM1 missiles from Jean Bart (that’s the video clip)
– 80x 100mm & many 20mm rounds from both destroyers

Despite all the above, the ship did not sink – possibly because there was no fuel, munitions & other flammable materials onboard.
Ultimately it had to be scuttled with explosives. This gives a good idea of the relative difficulty of sinking even a fairly small ship
(by today’s standards) with anything other than torpedoes or heavy bombs.
Here’s a series of pictures:

And in honor of its sterling Cold War service:
Indian Ocean (1960s?), in its original AA configuration

After a thorough anti-submarine refit (1980s?)

From here: http://www.alabordache.fr/marine/espacemarine/desarme/escorteurdescadre/vauquelin/photo.php