Controllable realisation of $\frac{s^4+1}{4s^4+2s^3+2s+1}$ is both controllable and observable?
Controllability is a property of $A,B$ and observability is a property of $A,C$  matrices. 

Transfer functions don't exhibit this property since if there is such a problem then you will not get a nonzero transfer function as there will be no possibility of control so nothing to transfer or nothing to observe. Hence, this is also a state space property. 

The main theorem says that a state space realization is minimal if and only if it is both observable and controllable. 

These representations allows one to quickly see, if exists, the unobservable and uncontrollable parts of the system. Such representations are obtained via the procedure called Kalman decomposition.

Here you have only constructed the controllable canonical form and already found out that the system is also observable. Thus the represantation is minimal.