Even with the Catholic Church, sure the Vatican has billions in the bank, the parish around the corner definitely doesn't.
"Non-profit" doesn't mean "doesn't have significant assets", it means "doesn't have shareholders or similar stakeholders with a claim on assets". (It is about not returning a profit to someone else, not about not making profits.)
My brother-in-law works in the construction supply industry. He isn't an owner or stockholder of his employer, he is just one of many non-managerial employees. His employer owns a holiday home and allows the employees to use it for vacation. (They have to apply in advance, and there is some system worked out on how to share it out – I think there is a waiting list, and also more senior people may get some priority.) When he takes his family on vacation to that holiday home, it is not a "return of profit", it is an employee benefit.
Historically, even relatively junior employees were often provided with accomodation. One of my favourite examples is a couple of unusual residences in Sydney, Australia, which are houses built into sewer vents [1], [2]. The government-owned water utility built these residences for the sewer workers who maintained the vents to live in (along with their families). That didn't make these sewer workers somehow recipients of a "profit distribution", nor did it make the government-owned water utility a "for-profit". (The houses still exist, but are now sold or rented-out to non-employees, since providing employees with residential accomodation is far less common than it used to be–and the sewer vents are still in use to ventilate the main sewers that run beneath them.)
[1] https://en.wikipedia.org/wiki/Corunna_Road_Sewer_Vent_and_Co...
[2] https://en.wikipedia.org/wiki/Premier_Street_Sewer_Vent_and_...
Yes, when the company owns it and keeps the house (potentially assigning it to your successor) when you leave the position, absolutely.