Transfer function and characteristic equation
I'm a bit confused by this since I thought that generally the
characteristic equation is the denominator of the transfer function.

If the example question had a current source (instead of a voltage source) and asked you to find the terminal voltage (as the output) then the TF would be inverted because it's V/I = impedance. In this case it would be the denominator that is the characteristic equation.
So, the answer depends on what you are trying to solve.