Airbnb will probably follow your law, but other platforms that do not will flourish. They will be difficult to regulate, given that they won't be in your jurisdiction.

> You could also make laws to limit the number of properties per person.

Many laws already exist that give unfavourable tax treatment to second and subsequent properties. They are widely avoided through family members and legal entity structures and a hand-wavy "map of relationships between people" isn't going to magically fix that. This would also have a fairly obvious negative effect: it would artificially limit supply in the rental market, driving up prices. A person who could have invested in multiple properties and supplied them to the rental market will instead supply at most one, and invest the rest of their money elsewhere.

There isn't some nefarious conspiracy preventing effective regulation in this area. It's just a legitimately hard problem.