Many if not most authors use the term “$n$-dimensional Euclidean space” as synonymous with “$n$-dimensional real space”, $\mathbb{R}^n$. Some, however chose to be more rigorous and use the term Euclidean space and the symbol $\mathbb{E}^n$ to specifically refer the metric real space equipped with the usual Euclidean distance function, whereas the general real space $\mathbb{R}^n$ is the set of real $n$-tuples, which is not necessarily equipped with additional structure such as a metric.
Considering this distinction, are $n$-dimensional manifolds locally homeomorphic to $\mathbb{E}^n$ or just to $\mathbb{R}^n$? In other words, do the codomains of the coordinate maps of charts in a manifold’s atlas have to be open subsets of a metric space with a distance function?
Thanks to the comments, and chewing on this a bit more, I now see the gap in my thinking on what may be a somewhat nitpicky point.
A homeomorphism is by definition a function between topological spaces. So when we say that every point in a manifold is an element of an open set which is homeomorphic to $\mathbb{R}^n$ we mean to an open set of the topological space “$\mathbb{R}^n$ with the standard topology”. The standard topology is induced by the Euclidean metric so this topological space is in fact the Euclidean metric space $\mathbb{E}^n$. So the shorthand description of a manifold as “locally Euclidean” is not as loose as I had thought. One does in fact need the Euclidean metric to induce a the standard topology on plain old $\mathbb{R}^n$, but this does not imply that the manifold itself is equipped with a metric.