A Euclidean space is an affine space over the reals such that the associated vector space is a finite-dimensional inner product over the real numbers.
https://en.m.wikipedia.org/wiki/Euclidean_space#Technical_definition
Is the theory of Euclidean space Euclidean geometry?
That theory of Euclidean space is not "Euclidean geometry". Here is a better wikipedia page for Euclidean geometry.
That link you posted is a great example of a bad wikipedia wiki page. The definition of Euclidean space in that link is a modern terminology that puts Euclid's name on a modern theory. It seems to be a standard definition, but not one I like personally. I would say that page you linked is confusing and conflating Euclidean geometry, Cartesian geometry, and the modern theory of vector spaces and affine spaces. Euclid did not even have coordinates.