Can this example prove that bode plot is not correct?
In standard form $$H(s)=\frac{\omega_n^2}{s^2+2\zeta\omega_ns+\omega_n^2}$$ your second order system $$H(s)=\frac{1}{(s^2+1)}$$ exhibits a damping ratio \$\zeta = 0\$ (no damping or an oscillator).

The main motivation (see the literature) for the construction of Bode Diagrams is that the terms in output corresponding to negative real part poles of a system will vanish at steady state (which not apply for oscillators). Even in these cases, the Bode diagram could still provide an interpretation for the presence of the two different frequency components on output. See the Bode Diagram below (magnitude only) for your system. At the frequency of 2 rad/s, the expected response occurs with a magnitude of 1/3. Now observe the peak, ie, the infinite gain at resonant frequency of 1 rad/s. Imagine that, at same time, there is an sinusoidal input of magnitude zero at this frequency (as it actually does), and that infinite gain will cause the response to assume the finite value which does not disappear. So you have both components in the output.