What rising_star said...
I worked in student housing and different functions within the property management industry throughout my undergrad...
Trust me, every company is different, every property within that company does things a little differently and every manager at every property has their own unique definition of what satisfies ID requirements.
On the one hand you have some companies who don't want a copy anywhere to reduce liability concerns for potential IDentity theft on the other extreme you might need to provide copies of your passport, visa, j-1/I-20, plus all of that for a guarantor/cosigner etc...
Ive never heard of any identity theft actually happening especially in student housing, and especially with international students... I wouldn't sweat providing them with copies of what they request.