One of the results from tensor calculus is that if all the components of a tensor are zero in one frame they are zero in all frames. Another result is that we can regard vectors as $(1,0)$ tensors.
In classical mechanics we have many vectors, and hence tensors, such as velocity and force. Consider the velocity $\mathbf{v}$ of a particle: if I transform via a Galilean boost to another frame which moves at speed $\mathbf{u}$ relative to the original frame, I find that the velocity measured in the other frame is
$$ \mathbf{v'}=\mathbf{v}- \mathbf{u}$$
Well if I transformed to a frame travelling at velocity $\mathbf{v}$, then I would find that the observed velocity would be $\mathbf{0}$. So what is going on here? I have a tensor which is non-zero in one frame and zero in another? Are vectors in physics truly tensorial objects?