  Anderson localization transitions are usually referred to as quantum phase
transitions from delocalized states to localized states in disordered systems.
Here we report an unconventional ``Anderson localization transition'' in
two-dimensional quantized quadrupole insulators. Such transitions are from
symmetry-protected topological corner states to disorder-induced normal
Anderson localized states that can be localized in the bulk, as well as at
corners and edges. We show that these localization-localization transitions
(transitions between two different localized states) can happen in both
Hermitian and non-Hermitian quantized quadrupole insulators and investigate
their criticality by finite-size scaling analysis of the corner density. The
scaling analysis suggests that the correlation length of the phase transition,
on the Anderson insulator side and near critical disorder $W_c$, diverges as
$\xi(W)\propto \exp[\alpha/\sqrt{|W-W_c|}]$, a typical feature of
Berezinskii-Kosterlitz-Thouless transitions. A map from the quantized
quadrupole model to the quantum two-dimensional $XY$ model motivates why the
localization-localization transitions are Berezinskii-Kosterlitz-Thouless type.
