Where is my error?
By definition a $4k$-dimensional manifold is quaternion-Kähler if its holonomy group is a subgroup of $(\mathrm{Sp}(k)\times\mathrm{Sp}(1))/\mathbb{Z}_2$ where $\mathrm{Sp}(1)=\mathrm{SU}(2)$ and $\mathbb{Z}_2$ is the intersection of the two factors in $\mathrm{Spin}(4k)$, namely a 2-element group.
For $k=1$ this holonomy group is isomorphic to $\mathrm{Spin}(4)$. All orientable (spin?) four-dimensional manifolds have a holonomy group contained in this, so all four-dimensional manifolds are quaternion-Kähler.
On the other hand wikipedia states that a quaternion-Kähler manifold is Einstein.
From all this I wrongly conclude that all four manifolds are Einstein.
You are correct that a $\text{Sp}(1) \cdot \text{Sp}(1) = SO(4)$, and so every orientable 4-manifold is quaternion-Kähler. Wikipedia means to demand $k \geq 2$. There is a clean proof of this fact due to Salamon that only really uses the representation theory of these groups; it is reproduced in Besse, "Einstein manifolds", p. 405 (14.41).
For what it's worth, Besse also make the claim (top of p. 403 in my edition) "As we shall see, the precise analog of quaternion-Kähler manifolds in dimension 4 are Einstein self-dual manifolds..." which are also studied in depth in chapter 13 of the Besse book.