Harvest Moon seems very similar to Animal Crossing, but can multiple players interact? Or would multiple players just have multiple independent single player experiences?
For the most part no, Console versions of Harvest Moon have not yet, to my knowledge; I've played 5 of them to date and found no evidence of the latest ones having multiplayer.
The Portable Console Harvest Moons have had some small multiplayer aspects; Harvest Moon GBC 2 and 3 had Trading features but no gameplay, but Harvest Moon: Tale of Two Towns has simultaneous multiplayer in the Multiplayer field. Sounds like the 3DS version has improved multiplayer aspects like Street Pass (and better resolution). Tale of Two Towns is by far the closest to what you're after.
Rune Factory 3, also for the DS, is an RPG style Spin off that has some of the core farming gameplay which also has a multiplayer dungeon. It would be the best/deepest multiplayer experience, but incorporates many RPG elements like combat as well as farming.