An MPS tensor $A^{[s]}$ is called *injective* if the map
$$
X\to\sum_s\mathrm{tr}[XA^{[s]}]\,\lvert s\rangle
$$
is injective.<sup>1</sup>

In particular, the example you give is *not* an injective tensor -- e.g., if you set $X=\begin{pmatrix}1 &0\\0&1\end{pmatrix}$, it is mapped to $0$. (In fact, you can already see this from the fact that $s$ can take only two values -- for the map to be injective it would have to take at least $4$ values.)



<hr>

<sup>1</sup> This map is sometimes denoted as $P(A)$ -- note that unlike what you say in your question, $P(A)$ is in general *not* a projector, even though in some important examples it is, and the "Projected" in "Projected Entangled Pair States", i.e., PEPS, refers to that.)</small>