Shrek
4th wall blurring: Aside from occaisional non-canon (though offically made) acknowledgement of the audience, Shrek himself is a send-up of more traditional fairy tale heroes on the meta level
Dominating: Often yells at Donkey and picks up and carries Fiona in the original movie
Marked Canon/Fanon divergence (or flanderization): Shrek in the movies is an anti-social recluse who eventually opens up to others, who posseses some strength but doesn't fight unless attacked first. However, around 2012-2017, many internet spaces used memes that tended to over-emphasize the gross-out elements of the character and sometimes outright portrayed him as a cosmically powerful, haunting abomination
Monster features: He's an ogre
Nonhuman: He's an ogre
Power: Is a good fighter due to him being an ogre, but doesn't seem to fight unless attacked first
Power of love: Shrek eventually becomes a better person due to his relationship with Fiona
Theme song: "Allstars" by Smash Mouth has become almost inextricably associated with Shrek
Unkempt man / grime: Shrek is unkempt with his frayed clothing and general demeanor, and the grime is the early 2000's gross-out humor
Distinctive voice: You know how he sounds
Memorable catchphrases: "What are you dong in my swamp"; "Onions have layers" etc
Patchy the Pirate (Spongebob)
4th wall blurring: Constantly breaks the fourth wall by speaking directly to the audience, and a lot of gags in his segments revolve around the fact that it's a set (such as pulling the parrot's puppeteer into the set). On a meta level, the character is a play on themed children's show hosts.
Chaoslord: His segments are particularly ridiculous even for a generally wacky cartoon like Spongebob. Everyone he interacts with seems to thik he's bizarre.
Distinctive voice: Patchy is portrayed by Tom Kenny, a professional voice-actor. Essentially, he's a live-action character with a cartoon's voice.
Dominating: A lot of gags revolve around him bullying the parrot for some reason
Eye imagery: Numerous gags bring attention to his eyepatch
Mysterious: A numer of fan theories have been made regarding who exactly the character is and what exactly his relation to the show and reality is
Purple: Patchy is always seen in a purple coat
Unkempt man: Looks and acts like a pirate
The Cat in the Hat
4th wall blurring: Especially in the 2003 live action movie where he often directly addresses the audience
Chaoslord: Definitely is this, does a lot of chaotic things in the name of "fun that is funny".
Distinctive voice: Tends to be given a pretty expressive voice
DILF/Father Figure: A big part of the 2003 movie's subtext is that the giant talking cat is a better father figure to the children than their soon-to-be stepfather.
Dominating: In general, he tends to steal the show the entire time he's on screen via force of personality
Mysterious: Like actually what the hell is he. what is going on. what is a three-handled moss-covered family credunza.
Nonhuman: He's a cat. In a hat.
Theme song: "Cat, Hat"
Well-dressed: Top hat, bow-tie, gloves
Technically antagonist: He's both the source of and solution to the conflict in the story.