No, except the Mechanical Engineering branch of MIT Manipal, it cannot be compared to one of the best private engineering colleges, BITS.
MIT Manipal may have better infrastructure than BITS Goa and Hyderabad but fails to give an edge in other aspects such as placements, coding culture, campus life, club culture, and brand name.
So it can be safely concluded that Manipal is improving, but it is yet to reach the level of BITS.