Yes, Manufacturing Engineering can be considered a sub-field of Mechanical Engineering. The reason for this is Manufacturing Engineering focuses on the use of tools and machines to impart form and function to raw materials or sub-components that comprise manufactured goods. This often requires knowledge of both mechanics and properties of materials, so Mechanical Engineering is a natural home for this kind of thing. There is some call for Electrical Engineering in this kind of work, but my experience has been that EEs are often placed in machine design or machine controls positions, while MechEs are often placed over a specific manufacturing process.
Strictly speaking, Industrial Engineering is different than Manufacturing Engineering, which is again different than Process Engineering, but in practice these terms can be used interchangeably because many engineers in these fields share these skillsets.
Industrial Engineering is related to fields like Operations Research, because it focuses on abstract, high-level representations of systems or processes. Thus things like time-studies, process flow, and systems simulations are performed by Industrial Engineers. However, some of these things are useful for a Manufacturing Engineer with responsibility for a specific manufacturing process, so there is considerable overlap.