In the story, these fantasy creatures are everyday animals that most people in the story already know. How can I introduce them in an natural way that instantly makes readers remember their names? The story is in comic book form, so that might open up a few more options.
The easiest way is to have someone say the name while looking at the animal and having a frame in your comic where the animal is the center. If you have an animal that plays the role that cats play in our real world for example you could have someone angrily say "Get the damn qutie from our table!" with a frame where this person is getting a water spray bottle while the animal is looking at the reader. This makes it clear what your character is referring to. The same would apply to other animals.
Give someone a reason to point out the name and look or point at the animal to make them the center of a frame. If you manage to make this an important thing your readers will surely remember it. Maybe there is an important document on the table. Or the qutie needs to get away from the water tank on the table with all the machies. Especially with a visual medium like comics this is quite easy for the reader to understand.
Have a "Cast of Characters" line up at the beginning showing all your characters with their names and include the creature(s) in some typical pose such as sitting on their owner's shoulder. You can also introduce dynamic by having an antagonistic character (if any) scowling or something at the creature on the other character's shoulder.
Comics often have a narrator adding content and context via boxes of text attached to the top or bottom of the frame. The narrator is allowed to break the fourth wall and talk directly to the audience, so introducing and explaining unusual animals doesn't have to burden the current scene or its characters.
Comic books give you a great opportunity to add footnotes.
In the Marvel and Dc tradition many story-lines cut across series. It is very common for there to be a rectangular box in a corner with a "*" that would help catch up the reader on what is going on, or what issue to read to learn the story, or follow a particular thread of the story. These footnotes are not something from the narrator, but are instead directly from the writer or editor. Comic readers are completely used to getting information in this way.
In exactly the same way you could use them to provide information on your creatures. They can come from the writer, or maybe from some character, a scientist or a wizard that we never see in panel giving us some fun insight on the creature. Maybe if the descriptions get particularity fun and witty, this character can sometimes get a page in a back of an issue where they could show what they do all day.