There doesn't exist a way to do this in the options of the obj importer it seems so what you could do is run a script that loops over the items in your scene and export each to a different file relative to the path your blend file is saved.

    import bpy
    
    # get the path where the blend file is located
    path = bpy.path.abspath('//')
    
    # loop through all the objects in the scene and export each to its own file based on its name and by selection
    for ob in bpy.context.scene.objects:
        bpy.context.scene.objects.active = bpy.data.objects[ob.name]
        ob.select = True
        
        # make sure that we only export meshes
        if ob.type == 'MESH':
            bpy.ops.export_scene.obj(filepath=str(path + ob.name + '.obj'), use_selection=True)
        ob.select = False