The average number of occupants doesn't change because everyone will still be living somewhere (excluding homelessness but we can ignore that). However, that does not mean cohabitation and second houses are irrelevant.

• Some people who were previously cohabiting (not renting) will buy homes
• Some people will buy second (or third, etc) houses.

Both of these scenarios are excluded from your example above. As a consequence of them, some renters will have to start cohabiting (or become homeless). The fact that there are more people who want to rent their (own) place then there are rental units will drive up rental prices.

https://pubs.aeaweb.org/doi/pdfplus/10.1257/aer.20181289

Here you go.

Using a 1994 law change, we exploit quasi-experimental variation in the assignment of rent control in San Francisco to study its impacts on tenants and landlords. Leveraging new data tracking individuals’ migration, we find rent control limits renters’ mobility by 20 percent and lowers displacement from San Francisco. Landlords treated by rent control reduce rental housing supplies by 15 percent by selling to owner-occupants and redeveloping buildings. Thus, while rent control prevents displacement of incumbent renters in the short run, the lost rental housing supply likely drove up market rents in the long run, ultimately undermining the goals of the law.