No. A visual example would be to imagine the line $\Bbb R$ embedded into $\Bbb R^2$ or $\Bbb R^3$ in some curvy way, e.g. as the graph of $\sin(x)$, etc. Write $\iota:\Bbb R\rightarrow \Bbb R^n$ for such an embedding, e.g. $\iota(x)=(x,\sin(x))$.
If you now view the distance between points of the line with respect to this higher dimensional space, you see that it depends on their actual position in space and not just their distance in the 1D-coordinate on the line. Formally, take the metric $d(a,b):=\|\iota(a)-\iota(b)\|$, which is far from translation invariant in the general case.