In Blender, object names and meshes names may be different, as in Cube2 and Cube.001. 
Is there a way to get the meshes name from the object name using Python?
Also,Is there only one meshes node per Object?
obj.name gives you the object name. obj.data.name gives you mesh name. And yes, AFAIK there can be only one mesh per object. However there can be multiple objects sharing the same mesh data block.
obj.namegives you the object name.obj.data.namegives you mesh name. And yes, AFAIK there can be only one mesh per object. However there can be multiple objects sharing the same mesh data block. $\endgroup$