The video was issued AQIM's al-Andalus media arm on Thursday and Lakome's Arabic and French sites wrote about it over the weekend.
The video was also front-page news in Moroccan newspapers on Monday.
Ahmed Bensiddik, a journalist for Lakome, confirmed Anouzla's detention in Rabat following the prosecutor's orders.
Morocco has rarely featured in al-Qaida's propaganda videos and the group has little organized presence in the North African kingdom, which is protected by heavily defended borders and a vigilant internal security force.
Terrorism in Morocco, including a blast at a tourist site in Marrakech in 2011, and bombs in Casablanca in 2003, has largely been done by small cells inspired by al-Qaida rather than actively controlled by the terror network.
The video echoes many of the criticisms by pro-democracy groups of the kingdom, including corruption and police brutality, but then goes on to fault Morocco for holding music festivals featuring openly gay performers such as Elton John in 2010.
Noting that many Moroccans attempt to immigrate to Europe on leaky boats in search of a better life, the video closes by urging them to take up jihad instead.
"Rise up young people to make your religion and the Quran victorious and save your nation because happiness is in the immigration to God and not on the boats of death," said Abdelmalek Droukdel, the leader of AQIM in an audio recording closing the video.