Depends. Is the realtime part of the draw for you? Magician’s Quest is definitely the closest equivalent, though I’ve also heard interesting things about recent release Cozy Grove.
Like the interacting with villagers and cozy vibe? Story of Seasons, Rune Factory, Stardew Valley, and cozy sims in that style are where to look. Of them, Stardew has an emphasis on customization, Rune Factory on character development, and for those who haven’t played, SoS is the mother of all farm sims. (Though I recommend Friends of Mineral Town or the 3DS games above newest title Pioneers of Olive Town. It’s fine, but… not their best offering.) Recent games also offer some nice character customization.
In ‘cozy sims in that style’, My Time At Portia features an emphasis on crafting that fans of New Horizons will likely enjoy. Littlewood does a fantastic cozy townbuilding game, with time passing only as you do actions like gathering, complimenting villagers (talking is a free action,) or catching fish and bugs - walking around costs no stamina and thus no time, which is nice because the accelerated time for these games sometimes does get a bit much. I’d say it probably comes closest to the fishing/insect hunting-interior design-social sim combo Animal Crossing focuses on. I’ve heard of a couple interesting fishing-specific sims - Moonglow Bay, Luna’s Fishing Garden - but haven’t tried any of them out yet in detail.
If you want a townbuilder with a more Minecraft-Meets-JRPG vibe, Dragon Quest Builders 2 is a delight I’ve been replaying lately, and the island theme of New Horizons and terraforming caused me to name my secondary character on my island after major character Malroth from this game. Also in RPG with vaguely similar vibes if not gameplay, the 3DS’s Fantasy Life. Less focus on collecting and social, but there’s plenty of crafting and decorating. (EDIT: I just remembered a very important feature in DQ Builders 2’s favor: Do you like the terrible puns when catching fish and bugs in Animal Crossing? This game has puns. Oh, how this game has puns. The central conceit is that the Children of Hargon, a villainous monster cult from the second Dragon Quest game, have a serious foothold on the island archipelago you’re living on for Reasons and have thus outlawed all acts of creation, especially that dastardly Building. I am currently on an island themed around farming, and have managed to convince the local pastor of the island of the magical ways of building and farming. Said pastor’s name is Al. Pastor Al. This happens more than once that I remember offhand, and I played this game three years ago. The puns are TERRIBLE and I cherish them.)
And if you want a really good dressup game and have a 3DS (or the Wii U, I guess) Style Savvy is a phenomenal boutique sim with very, very good fashion options. Three installments on the 3DS, Wii U has the original DS on the virtual console. Highly recommend.