The function $y = |x|$ is not a differentiable function from $\mathbb{R} \rightarrow \mathbb{R}$. But considering the graph of $y = |x|$ as a subspace of $\mathbb{R}^2$, we can endow this space with a smooth structure consisting of the single smooth chart which projects the graph down to the x-axis.
What I am confused about is how to reconcile these two seemingly inconsistent viewpoints. Can someone explain why $y = |x|$ is not a smooth function yet it can be made into a smooth manifold?
Yes, of course, one could endow this graph with a smooth structure. But then it is not a smooth submanifold of $\mathbb{R}^2$. In order for the graph to be a smooth submanifold, it needs a smooth structure which makes the topological embedding $$(\mathrm{graph})\hookrightarrow\mathbb{R}^2$$ a smooth embedding (that is to say, in this context, a smooth immersion). The smooth structure you suggest does not satisfy this condition, and, in fact, no other structure does either.