...repetitive tasks often struggle with retaining users: people abandon them because they feel bored, and boredom is simply lack of stimulation. By using positive stimuli like humor, movement, unique art, elements of game, and relatable characters we can make users feel a different way — more excited, less distracted, and ultimately happier.
From Design Emotional Interfaces
I don't think this type of element has a precise name, but perhaps analyzing all its components, a global idea may come up. There are some good articles about it in smashing magazine with some graphic and UI design concepts involved, but they all have the same goal:
The use of "mascots" in graphic design is nothing new and its meaning is quite well known: to bring the user closer to the product in a friendly way. There are several examples Bibendum, Energizer Bunny, etc.
Humor could potentially lead to a deeper emotional experience for the user, a key component for an effective UX.
Source: Incorporating Humor In Web Design
In this article there's a section within the types of humor that talks about Personification. The example interface of the question is almost a personification of a button.
Nearly every interface uses a form of animation. It’s the natural way to transition from one state to another. But animations in UI can serve a lot more purposes than signifying a change of state — they can help you direct attention and communicate what’s going on better than static visuals or copy ever could. The movement stimulates both visual and kinesthetic learning, which means users are more likely to stay focused and figure out how to use the thing.
Source: Designing emotional interfaces
Gamification is about using game-like setups to increase user motivation
Source: Relationship Between Gamification And Modern Persuasion
In particular I see gamification only in the animation since in this case there's no action after activating the button beyond interactivity: prompting to click on that button.
In summary, I think the animated mascot breaks the boredom of the interface with humor, stimulating the user to participate in the interactivity as a game.
Imagine the seek interface plain:
Putting all these concepts together, could we be talking about UI Mascots?