Confining / Constraining an object transform trajectory to a mesh

Does Unity have any built-in support for confining/constraining the movement of Object A to being only on the surface of a non-planar mesh Object B?

No, there's no built-in support for this. (The behavior you describe is very specific and isn't the sort of thing Unity would be likely to support directly.)

However, the topic has been discussed quite a bit both here and on the forums, and if you search for e.g. 'mario galaxy' you should find some relevant threads. (In short, a fairly common solution is to cast a ray downwards relative to the object, and if there's an intersection, move the object to the intersection point and align the object with the surface normal at that point.)