[TulliusCicero]

Owning as the only housing model doesn’t make sense. There’s plenty of reasons someone might prefer to rent instead, most of them having to do with not being in a position to put down roots. Which necessitates some kind of landlord.

[nitrogen]

Owning shares that could be transferred from house to house would make a lot more sense than renting.

[enord]

Money. The shares you are talking about already exist, and they're called "money". Of course, the relative value of shares wrt. the house will fluctuate over time and you'd have to account for this when transferring between houses, except this is calculated for you by the market when you just use money.

[TulliusCicero]

That sounds like it would be hideously complicated in practice.

Renting works fine if you have some basic regulations (which admittedly is the opposite of the case in the US).