Wonderland is Alice's imaginary Happy Place, based on the original Alice's Adventures in Wonderland.
This was already suggested on the YMMV page. In the episode "The Hatter Who Came to Dinner," Alice is shown reciting "The Walrus and the Carpenter," which obviously implies that the original Alice books exist in the world of the show. Also, the original Wonderland was All Just a Dream, so why shouldn't this one be too? We can assume that this '90s American Alice is a fan of the fictional Alice, so she's created her own fantasy of Wonderland that she imagines herself visiting. This would explain the modernized, Lighter and Softer characters, the fact that every episode's Aesop neatly corresponds to Alice's current problem at home, the existence of holidays like Christmas, Valentine's Day and Halloween in a fantasy world supposedly disconnected from Earth, and the fact that she sometimes seems to spend days or weeks at a time in Wonderland, yet no time seems to have passed when she gets home. Alternatively, this Alice and her friends are descendants of the original Alice and Wonderland characters.
This presumes an AU of the original Alice in which Wonderland wasn't All Just a Dream. After the original Alice grew up, she and her family moved to America. The Alice of this series is her great-great-great granddaughter, named after her, and the mirror she walks through to get to Wonderland is the same one the original Alice climbed through in the second book – a family heirloom, which at some point was converted from a mantlepiece mirror into a more practical floor mirror. The Wonderland characters are also descended from the originals, and their increased friendliness and racial diversity are the result of Wonderland's various social and cultural reforms across the decades; from their perspective, the original characters' rudeness and meanness was 19th century Values Dissonance. In "That's All, Jokes," the Cheshire Cat was the one who played the prank on the Queen.
We never find out who did it. But notice that the Cat isn't among the characters who assemble at the palace in the end, and none of the characters who do come confess to the prank, despite pressure from the Queen. Also, the prank happens outdoors, yet it involves things showering down onto the Queen from overhead. There's no ceiling space where all the streamers, rubber chickens, etc., could have been hidden. But the Cat could have been floating overhead, invisible, and dropped everything onto the Queen at the right time. Pinniped is the Carpenter.
The original poem in Through the Looking-Glass that features the Walrus is, of course "The Walrus and the Carpenter." The two are inseparable. Ergo, the best friend of this version of the Walrus is also the Carpenter. The poem never says that the Carpenter is visible, after all, and why should being invisible stop Pinniped from practicing carpentry?