Slightly pedantic, but AC just means football club. If you want to shorten AC Milan's name just Milan is fine, people will know what you mean

They're going to build a new stadium, now owned by both

The San Siro has been for a long ass time the stadium of the city of Milan, from back when football wasn't professional still. So they kept continuously the ownership. Bureaucracy, a hellbent city administration, etc but now the situation of San Siro got so bad, and the city ownership of stadiums is so disliked, and the city administration got a bit more open minded, that they approved plans to the building of a stadium in another part of the city. All self financed by AC Milan and Inter, but they'll have the full ownership of the stadium. Will be called cathedral or something, kinda pretty stadium.