The collective warned that Paris-based Early Flicker's registration of the image of a headless suited figure for use in France "will not go unpunished".
The company has also registered Anonymous's slogan: "We are Anonymous, We do not forgive, We do not forget. Expect us."
In a YouTube video, the hacktivists warned that their name will not be the "whore of the world".
The group confirmed that Early Flicker has registered the Anonymous logo and slogan through the National Institute of Industrial and Property (INPI), meaning it holds the rights in France
"Their arrogance and ignorance of what they have done will not go unpunished," warned the group.
Anonymous said that they "will take down" any Early Flicker operations on the internet, and this campaign "will not stop until the registration has been dropped and a public apology has been made".
"The name of Anonymous will not be the whore of the world," they added.
The image of a headless, suited figure has become synonymous with the group since it burst into popular consciousness with a series of hacking attacks on high-profile websites.
Members of the group, including the person depicted in the video, are also frequently seen wearing masks depicting the character Guy Fawkes from popular graphic novel, V for Vendetta.
But Early Flicker, also known as EFlicker, has already started selling T-shirts displaying the group's logo on its website, and to add insult to injury for the group, they are listed in the "geek" section.
The firm has not responded publicly to Anonymous's demands.
The INPI granted the trademark after judging that the "conditions seem to be fulfilled as the mark is registered and it does not seem to have made use".
Anonymous has now launched Operation AnonTrademark, which aims to get the trademark revoked.
The group has previously targeted certain websites with Distributed Denial of Service (DDoS) attacks, which overload an online presence with requests until it is unusable.