A possible characterization of Euclidean geometry via the curvature of the Median-submanifold Is there a  Riemannian metric $g$ on $\mathbb{R}^2$ with inducing  distance
$d$ which is  not isometric  to the  standard metric  but satisfy the  property  quoted bellow?

For  every two  distinct  points  $p,q\in \mathbb{R}^2$, the  locus  of  all  points $z$  with $d(z,p)=d(z,q)$ is  a geodesic.

As a higher dimensional version:
Is there a  Riemannian metric on $\mathbb{R}^3$ which is  not isometric  to the  standard metric  but satisfy the  property  quoted bellow?


For  every two   distinct points $p,q\in \mathbb{R}^3$, the  locus  of  all  points $z$  with $d(z,p)=d(z,q)$ is  a 2 dimensional  submanifold with constant sectional curvature?


Edit:
According to the comment of  Willie  Wong, I realized that  the  previous version of the  question should be  reconstructed. So  I  present the  question as follows:

Assume  that we  have  a complete Riemannian  metric  on $\mathbb{R}^n$  which satisfies the  following property:  For  every $2$ points $p,q$, the  locus  of  all points $\{z\mid d(z,p)=d(z,q)\}$ is  a  codimension- $1$  smooth submanifold  which is  a totally geodesic  submanifold. Does  this  imply that the  metric  is  isometric  to either Hyperbolic  space  or  the  Euclidean space?

 A: The answer to your question is in the affirmative, even in the more general setting of general (pseudo)-Riemannian manifolds.
In the Riemannian case this is a theorem due to Busemann. Consider a Riemannian manifold $(M,g)$. The set $\Sigma_{p,q}$ consisting of all points equidistant from $p$ and $q$ on a Riemannian manifold is called the bisector of $\{p,q\}$. Busemann's theorem states that if $\Sigma_{p,q}$ are totally geodesic for any $\{p,q\}$, then $(M,g)$ has constant sectional curvature.
A generalization was proven by Beem for pseudo-Riemannian manifolds. The definition of "distance" and "bisector" requires a bit of care. Both are only defined within "simple convex neighborhoods", and the bisector is only a submanifold when $p$ and $q$ is not joined by a null geodesic. It turns out that this "local" distance function and "local" bisector, for pairs of points that are time-like or space-like separated, is enough to imply Busemann's theorem.

References
Busemann, H., The geometry of geodesics, Pure and applied Mathematics, 6. New York: Academic Press, Inc. X, 422 p. (1955). ZBL0112.37002.
Beem, John K., Pseudo-Riemannian manifolds with totally geodesic bisectors, Proc. Am. Math. Soc. 49, 212-215 (1975). ZBL0301.53026.
