Can I request a Ph.D. in a different department having done research in another? I am a Ph.D. student in mechanical engineering and almost done with my thesis. My bachelors was in Physics. Although the initial research plan fit into mechanical engineering - vibrations, thanks to the freedom my supervisor gave me, I gradually moved into dynamical systems theory, complex systems and networks, computing, neural networks, optimization, etc.
Looking at my thesis, it is a stretch to call myself a Ph.D. in mechanical engineering.
It would make more sense to obtain a Ph.D. in Physics or Applied Mathematics rather than mechanical engineering

Has anyone faced such a situation? Can I request the university for this? If yes, how would I go about this?
Talk to your supervisor. This probably differs from country to country (maybe federal state to federal state) and university to university. 

However my experience is that people either want to know if I have a PhD or not, or, if they need more details, then they want to know the topic of my dissertation. The intermediate level of the discipline has never come up. So I would not worry too much about it.