Quick question: Direct sum of zero-dimensional subschemes supported at the same point

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For a torsion free sheaf $E$ on a surface $X$ we have

$0\rightarrow E\rightarrow E^{**}\rightarrow Z\rightarrow 0$

where $Z$ is a zero dimensional subscheme of $X$.

Let $p\in\mathbb{P}^2$ be a closed point. Let $\mathcal{I}_p$ be the corresponding ideal sheaf.

We have $(\mathcal{I}_p^{\oplus 2})^{**}=\mathcal{O}^{\oplus 2}$. Hence $\mathcal{O}^{\oplus 2}/\mathcal{I}_p^{\oplus 2}$ is a zero dimensional subscheme of $\mathbb{P}^2$.

But $\mathcal{O}^{\oplus 2}/\mathcal{I}_p^{\oplus 2}=\mathcal{O}_p^{\oplus 2}$ - not a subscheme. Where did this go wrong?