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][1].


  [1]: https://doi.org/10.1016/j.aim.2020.107131