To get uniqueness of the SDF, you need complete markets. In a world with $n$ possible outcomes, complete markets would require $n$ linearly independent securities. Without complete markets, the SDF is not unique.
Simple Example
Imagine there are two states of the world and I represent probability measures and random variables as two-dimensional vectors. Let:
$$ \mathbf{p} = \begin{bmatrix}\frac{1}{2} \\ \frac{1}{2} \end{bmatrix} \quad \mathbf{x} = \begin{bmatrix}3 \\ 1 \end{bmatrix} $$
$$ \mathbf{m}^{(a)} = \begin{bmatrix} \frac{1}{3} \\ 1 \end{bmatrix} \quad \quad \mathbf{m}^{(b)} = \begin{bmatrix} \frac{1}{2} \\ \frac{1}{2}\end{bmatrix}$$
Let's imagine the price of security $x$ is 1. Both $\mathbf{m}^{(a)}$ and $\mathbf{m}^{(b)}$ correctly price the security since $\sum_i p_i m^{(a)}_i x_i = \sum_i p_i m^{(b)}_i x_i = 1$.