The term is 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.