Let $(H, \langle\cdot, \cdot\rangle)$ be a Hilbert space and $P: H \to H$. In this answer, @gerw said that if $$\forall (x,y) \in H^2:\langle Px, y \rangle = \langle x, Py \rangle,$$ then $P$ is linear. Because I'm trying to prove it by myself, I've not read his/her solution.
On the other hand, From this Wikipedia link about self-adjoint operator,
In mathematics, a self-adjoint operator (or Hermitian operator) on a finite-dimensional complex vector space $V$ with inner product $\langle\cdot, \cdot\rangle$ is a linear map $A$ (from $V$ to itself) that is its own adjoint:$\langle A v, w\rangle=\langle v, A w\rangle$ for all vectors $v$ and $w$.
From the paragraph, I got that self-adjoint operator is not necessarily linear. If it was, they would not said "linear map $A$ that is its own adjoint". Could you please reconcile this difference?