The bag— basket shaped , with a ruffled trim along the top and snap-removable straps—can be seen wherever there are well-dressed moms pushing strollers. They're perched outside daycares and music classes; they're joining on daily errands and meet-ups with fellow moms at the park. At a new Upper East Side outpost of the coffee chain Fellini on a recent Thursday morning, there are two sets of chatting parents at the walk-up window, two gingham stroller basket bags perched beneath the fresh iced lattes in their cup holders. Any tote or backpack can serve as a stroller bag, with ample storage space and a wipe-down interior. But for a growing network of moms from New York to Paris, it's this bag, MiniCoton's Basket Bag , that's the only option they want dangling from the handlebars of their Bugaboos and Nunas.…