If there is an isomorphism $\mathrm{Hom}(y C, P) \cong P (C)$ natural in $P$ and $C$, then by restricting to the representable functors one obtains an automorphism of the identity functor of the category $C$ comes from. There are categories for which the identity functor has a non-trivial automorphism group, e.g. $\mathbf{Ab}$, and also categories for which the identity functor has a trivial automorphism group, e.g. $\mathbf{Set}$.

In more detail: let $P = y D$. Then $\mathrm{Hom}(y C, y D) \cong \mathcal{T}(C, D)$, and if we compose with the *standard* Yoneda isomorphism we get an isomorphism $y D \cong y D$, natural in $D$. The Yoneda embedding $y : \mathcal{T} \to [\mathcal{T}^\mathrm{op}, \mathbf{Set}]$ is fully faithful, so this determines an automorphism of the identity functor on $\mathcal{T}$.

Conversely, suppose $\theta : \mathrm{id}_{\mathcal{T}} \Rightarrow \mathrm{id}_{\mathcal{T}}$ is an automorphism. Then we can compose it with the Yoneda embedding and use functoriality of $\mathrm{Hom}$ to get a new natural bijection $\mathrm{Hom}(y C, P) \cong P (C)$: explicitly, it is the map defined by evaluating $P$ at $\theta_C$.