It's not going to help them find a place to live though: they'll be stuck in hotels the entire time they're here.
Eh, Tokyo has plenty of monthly apartment rentals that are effectively corporate rentals that you can do without a residence card.
(I used to do this before having an actual visa there)
It's functionally better than a hotel, insofar as it doesn't read or act as one.
The court was willing to stamp and sign that I have that passport and it matches me, so it is probably good enough for most police if I offer to show them the original at home.
That being said having a photo of your passport and relevant stamps is good advice, but only to make it slightly easier to deal with all the hassel that comes if you do lose your passport.
Note that this doesn't include your passport stamp pages but Japan hasn't issued passport stamps for several years now and they just look up your passport in a registry instead. So for that purpose, a true copy should be effectively the same thing.
That's where it comes down to what is essentially a technicality but given they carry an embossed seal and signature with the words "TRUE COPY" on them, they look very official and officials are very rarely going to push back on it even if whether they are to be treated as a full substitute for a passport (for identification purposes) is technically a grey area.
Unless you are involved with a particular interaction, they just want to check your visa status. A passport photo/stamp will do if their system is digitized. Arresting someone (except for the US where the police likes to arrest people) is a major hassle.