For the atoms $\phi = P(x, y)$ and $\gamma = P(x, y)$, provide a unifier, if it exists.
Obviously a unifer is not needed for these two atoms, but does one exist?
Can a substitution $S = \varnothing $? If so, would that substitution unify the two atoms? $S(\phi) = P(x,y) = S(\gamma) $
What about $ S = \{(z/x)\}$? $S(\phi) = P(x,y) = S(\gamma) $
A substitution $\sigma$ is a unifier for an expression $\phi$ and an expression $\gamma$ if and only if $\phi \sigma=\gamma\sigma$.
A substitution is a finite mapping of variables to terms. I will write substitutions as sets of replacement rules, like the one shown below. In each rule, the variable to which the arrow is pointing is to be replaced by the term from which the arrow is pointing. In this case, $x$ is to be replaced by $z$, and $y$ is to be replaced by $w$.
\begin{align} \sigma=\{x\leftarrow z,y\leftarrow w\} \end{align}
Consider the substitution $\sigma$, trivially, this makes the two expressions $\phi$ and $\gamma$ the same.
\begin{align} \phi\sigma&=P(x,y)\{x\leftarrow z,y\leftarrow w\}\\ &=P(z,w)\\ \gamma\sigma&=P(x,y)\{x\leftarrow z,y\leftarrow w\}\\ &=P(z,w) \end{align}
Your substitution $\{x\leftarrow z\}$ is a unifier for the expressions $\phi$ and $\gamma$. Application of the empty substitution will also unify the two expressions.