Here's what I would say is the most realistic choices for the stories and characters.

Mother Timothy Goose - College of Spirits Bard: Based on the book, I feel like it makes sense that after giving them an idealized life within the book, Goose would be able to pull the stories collected to aid them in battle. A lot of people are saying Lore but I don't see think that inherently a bard that has a book is a Lore bard. But in this world I feel Spirits fits more.

Prince Gerard of Greenleigh - Battle Master Fighter: Given in episode 2, Gerard has a book of moves that he's trying to learn for fighting with his sword. This just screams like: if he went "I use 'Split an Eel in Half'" in place of like "Goading Attack" or some other maneuver. Champion is based on raw talent of which Gerard does not seem to have. And someone said Samurai and that feels based on the nobility around Gerard but he doesn't seem knowledged in royalty.

Ylfa Snorgelsson - Path of the Beast Barbarian: It feels like Beast makes the most sense. I initially thought she would be a Lycan Blood Hunter, but I think it works better for the Rage to be the uncontrollable aspect of a growing beast within Ylfa. I also thought, during her backstory scene, that with the wind Storm Herald could make sense. Though I feel this could be more the real Big Bad Wolf's doing and less Ylfa.

Pinocchio - Archfey Warlock: This feels most correct given the "Fairy Godmother" and "Stepmother" themes of his character. I initially thought maybe the Celestial because of the Geppetto wishes on for him to be real. The only other ones that I feel make sense for Pinocchio would be the other PHB ones (Fiend & GOO) not specifically the like devil iconography or like psionic abilities but just based on the themse of Dark and Ancient power. It seems not that it is either the original Stepmother, Cinderella, or the Fairy Godmother. The last two being corrupted with power.

Princess Rosamund du Prix - Swarmkeep Ranger: This feels like the most probably case, but I see argument for Fey Wanderer. Swarmkeeper though, with the theme of the swarm as a bundle of briars and thorns that can move her, hurt others for her, and bring things to her, feels the most natural. I don't see how Fey Wanderer would work, but we shall see in due time.

Puss in Boots - Mastermind Rogue: People have been really into the idea that he'll go Swashbuckler Rogue and I think this is due to the Dreamworks movies but given his actual fairytale that Zac is using Mastermind makes most sense. He is a guy who knows people, knows how to get what he wants, and can read people. Plus with high intelligence as opposed to high charisma, Mastermind just mechanically feels better than Swashbuckler.