I think it worth mentioning that precisely the notion described in the question is given in Cockett–Koslowski–Seely's Morphisms and modules for poly-bicategories, where it is called a multi-bicategory. A one-object multi-bicategory is a multicategory. As noted in the comments for Simon Henry's answer, Leinster's fc-multicategories are strictly more general: they are to multi-bicategories what pseudo-double categories are to bicategories. A multi-bicategory is precisely an fc-multicategory whose vertical cells are all identities.

I think it worth mentioning that precisely the notion described in the question is given in Cockett–Koslowski–Seely's Morphisms and modules for poly-bicategories, where it is called a multi-bicategory. A one-object multi-bicategory is a multicategory. As noted in the comments for Simon Henry's answer, Leinster's fc-multicategories are strictly more general: they are to multi-bicategories what pseudo-double categories are to bicategories. A multi-bicategory is precisely an fc-multicategory whose vertical cells are all identities.

In the terminology of Cruttwell and Shulman, the structure you describe would be called a virtual bicategory.