they recycle the bathwater to use in the laundry
No worse, really, than suds-saving, esp. when you consider that they enter their bath water very clean anyway.
If they do this, it would probably be to save on the energy used to heat the bath water. So the hot water could be used for a hot/warm wash cycle, followed by fresh water for rinsing. Or even bath water for rinsing, since they don't put any soap in their bathwater, either, but maybe not. Water is plentiful in Japan, energy is very expensive.