This is the first sentence of a conditional I need to construct a false interpretation for. So I need to find an interpretation that makes this sentence true. However, it looks to me as if this can never be the case.
Since both variables $x, y$ are universally quantified, I would have to extend the predicate P to include all elements of the domain of the model in order to make the sentence $P(x) \wedge P(y)$ true.
So wouldn't there always be a case where $x = y$? I.E this sentence can never be true? Any insight is greatly appreciated. Thanks in advance.
If the domain of discourse is not allowed to be empty, then the sentence is unsatisfiable for exactly the reason you gave, namely that (101) is false.
$$ \forall x \forall y . x \neq y \tag{101} $$
Let $c$ be an element of the domain. Pick $x = c$ and $y = c$ , now (101) is false. The further $P(x) \land$ and $P(y) \land$ do not affect the analysis because an expression headed by $\land$ is false if either of its arguments is false.
If the domain of discourse is allowed to be empty, then the sentence is vacuously true because the outermost quantifier is universal. All sentences of the form (102) (with $\varphi_x$ being an arbitrary sentence that potentially includes $x$) are true because there body of quantifier is never "evaluated" and doesn't get a chance to influence the truth value of the expression as a whole.
$$ \forall x . \varphi_x \tag{102}$$
Roughly speaking, if you think of (102) as a game, then the opponent makes a move and gets to pick an item from the domain. The domain is empty though, so the opponent can't move and therefore loses.