The term that's usually used is hereditarily equivalent.
As Will Brian pointed out, the pseudo-arc has this property. It is indecomposable, i.e., it is not the union of two of its proper subcontinua. G. W. Henderson showed that a hereditarily equivalent decomposable continuum is an arc (homeomorphic to $[0,1]$).
It is still a big open question as to whether the arc and the pseudo-arc are the only hereditarily equivalent metric continua.
Edit July 2020: The problem alluded to above has been solved for planar continua. The pseudo-arc is the only "minimal" (i.e. hereditarily equivalent) plane continuum other than the arc: https://doi.org/10.1016/j.aim.2020.107131.