At the last Italian census just over 1% of Italian families still lived in multi-generational households.(Istat 2016)
In Britain it's 1.2 million people who live in multi-generational households. (Yougov survey 2019)
Or, almost exactly the same.
Italy didn't take it seriously enough initially, no. Only the hotspots were locked down and then there was a gradual extension of that. Schools in and near to the hotspots were closed totally as soon as the first cases appeared.Schools nationally on the 4th March.
On the 7th March the fact that Lombardia was locking down on the 8th caused a mass exodus of people to the south. (They were not "sent" as a pp has said, they fled)
In one Italian region (Puglia) the governor said on Thursday that 40% of those in critical condition had had contact with someone leaving Lombardia on the 7th. Estimates are that 200,000 left.
Those people were what persuaded the govt to take further measures 3 days later and place the whole of Italy on lockdown. That too has also been made more stringent as people didn't think it applied to them. Individual town councils are also imposing their own ever stricter rules (no dog walking, shops on restricted hours etc) Lockdown now is being policed very very strictly, sure there are twats who'll try and meet their mate, but they are very isolated incidents.
Lombardia is densely populated and industrialised. There are many tourist destinations nearby. 5 foreign borders in the north of Italy.
Patient 1 was a superspreader. Patient 0 was German and had left the country so it was a while before the real alarm was raised.
Etc etc.
But ultimately, no, not because Italians all live together (they don't, as the Istat figures show) Not because they're old, (Italians live so long because they're healthy by and large) Not because their health service is rubbish because it's excellent.
But because they didn't take it seriously and the initial measures were not far-reaching enough.
They are now, and peak contagion is expected in a few weeks with peak deaths rate following shortly.