Are thoughts transferred along with a brain during a brain transplantation? If I'm transplanted with, e.g., Stephen Hawking's brain, would I start to think like him, or would I remain to think like me? On which, "thinking" ability depends , is it the physical structure of brain or any non physical thing?
Short answer
Human brain transplants have never been performed, so any answers to what will happen are mere speculations.

Background
Whole-head transplants have been performed on animals, the first reports go back to the 1970's (Canavero, 2013). The practice is still pursued today (e.g.., Ren et al., 2015)). Up until now, however, it has not been done in humans, and likely it never will be (source: The Guardian). This, because of medical limitations (Canavero, 2013) as well as ethical considerations (Wolpe, 217)

The biggest problem medically is the re-connection of the thousands of spinal neurons linking the brain to the rest of the body (Canavero, 2013). Further, immune responses and immediate cell death due to oxygen depletion may all further hamper the process. Nonetheless, certain groups envision it will be at one time possible in humans (Canavero, 2013).

The same limitations as with head transplants discussed above apply for a brain transplant. The thing is that with a whole-head transplant the major arteries can be quickly connected to feed the brain. With a brain transplant this also becomes a major bottle neck, making it even more difficult. Transplanting a head can be done with relative ease by sacrificing the donor and straight away stitch it to the beheaded recipient. A brain transplant needs even more steps. The big advantage of a head transplant is also that  the brain, being within its own cephalon, still connects to the cranial nerves (White et al, 1975).  

References
- Canavero, Surg Neurol Int (2013); 4(S1): S335–S42
- Ren et al., CNS Neurosci Therapeutics (2015); 21(8): 615-8
- White et al. Resuscitation 1975;4,197-210
- Wolpe, AJOB Neurosci (2017); 8(4)

Further reading
- Is brain transplantation possible?