It does not mean on the other hand.
We use on one hand / on the other hand when weighing up two sides of an equation, an example of which could be talking about the pros and cons of doing something.
A 'spectrum' is a way to refer to a gradation of change, the most illustrative example being the colours of the rainbow, where red is at one end and blue is at the other.
The student's phrase on the opposite side of the spectrum is incorrect for several reasons. You should not use it.