The answer is no. It's not even true for ellipses in 2D. In 2D, the derived shape is called an "offset curve", or, in older texts, a "parallel curve". In 3D, the term "offset surface" is used. You can find a good introduction to the topic in two papers by Farouki and Neff: Analytic properties of plane offset curves & Algebraic properties of plane offset curves, Computer Aided Geometric Design 7 (1990), 83-99 & 101-127. This sort of offsetting is important in CAD/CAM, so it appears frequently in the literature.
I'm not aware of any specific known results about the quality of the approximation. Clearly it's pretty good when the $\epsilon$ values are small and very bad if any $\epsilon$ value is larger than the minimum radius of curvature of the original ellipse. In 2D, we have two curves with known (and fairly simple) equations, so figuring out the error distance between them shouldn't be too hard. I expect a few minutes with Mathematica would yield some insight.
Edit: I found a somewhat related paper, again by Rida Farouki. Conic Approximation of Conic Offsets, J. Symbolic Computation (1997) 23, 301-313. Not exactly what you were asking about, since it deals with general conics rather than ellipses, specifically, but may be helpful.
I don't know anything about $d$ dimensions (unless $d$ happens to be 2 or 3). Offsetting problems are hard enough even in the low-dimension cases, so I've never gone beyond.