The Cayley graph of a finitely generated countable group is a metric space with respect to the word metric. One includes both edges and vertices in the geometric realization (after all, it's called a graph, not a discrete point set). If the group is nontrivial, the graph is regular of constant finite valence, so small neighborhoods of any point are isometric to line segments or stars. If the group is nontrivial, the Hausdorff dimension of the graph is 1, and this is independent of the choice of generating set. Any countable Cayley graph can be embedded (non-isometrically) in $mathbb{R}^3$ using an enumeration of vertices by natural numbers, e.g., sending the $n$th vertex to $(n,n^2,n^3)$, and taking edges to be straight lines.
The free group on 2 letters can be embedded (non-isometrically) in $mathbb{R}^2$ as a fractal, as shown in the linked image you gave. The Hausdorff dimension of the image in general strongly depends on the choice of embedding. For example, the Cayley graph of $mathbb{Z}$ with generator $1$ is isometric to the real line, but you may be able to embed it in a larger space using some kind of Brownian motion, where the image will have Hausdorff dimension 2 almost surely.
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