According to Wikipedia, we have that the Hodge star is defined so that $$\alpha\wedge *\beta=\langle A,B\rangle\omega,$$ so it seems that in 4-dimensional space with basis $x,y,z,w$ and $\omega=x\wedge y\wedge z\wedge w$, we should have $$*(x\wedge y)=z\wedge w$$ and also $$*(z\wedge w)=x\wedge y$$ because $\langle x\wedge y, x\wedge y\rangle=1$ and $\langle w\wedge z, w\wedge z\rangle=1.$
However, again according to Wikipedia, we have $$**(\eta)=-1^{k(n-k)}s\eta,$$ where $s=1$ if $\langle\cdot,\cdot\rangle$ is an inner product. So, we should have $$**(x\wedge y)=-x\wedge y.$$
Which step am I making a mistake in?
In the formula $**(\eta)=(-1)^{k(n-k)}s\eta,$ $n$ is the dimension of the vector space and $k$ is the degree of $\eta$. So in your case with $\eta=x\wedge y$, $n=4$ and $k=2$ so $(-1)^{k(n-k)}$ is $1$, not $-1$.