I recall my professor mentioning that Yoneda's Lemma can be used to show that limits of a functor are unique up to isomorphism. Here is my attempt:
Let $F:J\to\mathcal{C}$ be a functor and let $X$ and $Y$ be limits for the functor. Then for every object $Z\in C$, we have a bijection $\text{Hom}_\mathcal{C}(Z,X)\cong \text{Cone}(Z,F)=\text{Hom}_{\text{Psh}(\mathcal{C})}(\Delta(Z),F)$ where $\text{Psh}(\mathcal{C})=\text{Fun}(\mathcal{C}^{op},\textbf{Set})$ and $\Delta(Z):J\to\mathcal{C}$ is the constant functor whose image is $Z$. These bijections together constitute a natural isomorphism from $\text{Hom}_\mathcal{C}(\cdot\ ,X)$ to $\text{Cone}(\cdot\ ,F)$ as functors from $\mathcal{C}^{op}$ to $\textbf{Set}$. By the same reasoning, $\text{Hom}_\mathcal{C}(\cdot\ ,Y)\cong\text{Cone}(\cdot\ ,F)$ so that $\text{Hom}_\mathcal{C}(\cdot\ ,X)\cong\text{Hom}_\mathcal{C}(\cdot\ ,Y)$. However, because the Yoneda embedding is fully faithful (a consequence of the Yoneda Lemma), this implies $X\cong Y$ in $\mathcal{C}$.
Working through the isomorphisms, one can see that the isomorphism from $X$ to $Y$ obtained here is actually the morphism induced by the universality of $Y$, so the limit is actually unique up to a canonical isomorphism with respect to the limiting cone. Is all my reasoning correct?
Your reasoning is correct, but you can stop after the first natural isomorphism. You are stating that since $X$ is a limit it represents the functor sending an object to the set of cones with that object as its summit. Symmetrically, $Y$ also represents this functor. By the Yoneda lemma, these objects must be isomorphic through an isomorphism respecting the representations.