Normally a piece of property has one owner. This means that if there is a dispute about how to use the property, it can easily be resolved: defer to the owner. By giving everyone a fallback position which they can agree to, this helps make society more peaceful, and helps people get along.
Even in a family or in personal relationships, property ownership is important. If I want to read a book, and my friend wants to read a book at the same time, we'll often decide who gets to read it by who owns the book. And when it comes to my computer, it's important that I have full control over it, and never have to debate or defend my use of it if I don't want to.
Sometimes people share property. They both own it. Perhaps they split the price, or they are siblings and their parents gave it to both of them. The danger of sharing is that there's no clear procedure for what to do when people disagree about how to use the property. It's best if a procedure is decided in advance, and known to all parties. That helps avoid creating a biased procedure, or an unexpected disappointment.
Large companies like Apple or Coca Cola appear to have many owners. Are they making the mistake of sharing? What's going on is that the company was divided into many small pieces (shares of stock), and each piece has exactly one owner. When the company was divided, a well defined procedure was created for how the company's decisions would be made, and how disputes would be settled. Because nothing is actually shared, and there is a mechanism for resolving disagreements, there is no problem.
The problem with shared property is that if people disagree about it — perhaps they both want to use it right now — there is no clear way to decide what to do. That means they have unlimited potential to fight. A common response is to say, "Never mind, I don't care, it's not worth the bother". That is good for avoiding a fight, but what's the point of having something if you're going to take that attitude? And when that attitude is used, it's often the more reasonable person who loses out; it's bad to create incentives for people to be stubborn or unreasonable.
Disagreements between people are common and should be expected. Buying something together and expecting never to disagree about its use is naive. Our society has built up complex knowledge-containing traditions for resolving disputes. By using these traditions, serious disagreements become more uncommon, but still fairly common. When people abandon these traditions, for example by sharing ownership of something, they are giving up the use of this important, problem-solving knowledge. The expected result of not using knowledge of how to solve problems, and trying to make something up from scratch to replace it, is to fail. Most new ideas don't work. The only reason our existing traditions work well is because many people have improved them.
Sharing is bad because it's bad to ignore or reject important problem solving traditions. Sharing is bad because we should not risk fighting, especially with our friends or family. What person, who realizes disagreements are common, would want to create a situation where disagreements could easily become fights?
There is a second kind of sharing which is to temporarily lend someone your property. This is often done with no clear agreement about when it will be returned, so it has dangers similar to the above. However, when push comes to shove there is still an owner, and the property will generally be returned prior to a serious fight taking place.
Lending is most common in two situations. First, adults often lend things that aren't important to them (e.g. a book, which they already read, and which they could just buy a new copy of if worst came to worst, so it's no big deal no matter what happens). There's nothing wrong with this since the downside is limited. Second, children are often forced by their parents to share their toys and other precious possessions. This has a much more serious potential downside because the child has few things, and cares about them, and doesn't have money to replace them.
Often children are made to share things they want to use right now. But why? That is anti-traditional. The property tradition says the owner should use his stuff when he wants to, and if he lets others use it that should only be at his convenience. What's the point of having property if you can't use it when you want to? I think what's going on here is the parent is the real owner of the property, and the child owns nothing. When the parent requires the child share, that is the parent using the property in the way the parent wants to: the parenting is "teaching his child a lesson" by making him unhappy. The parent is trying to force an altruistic, self-sacrificing morality on his child (one he does not practice in his own sharing decisions: he does not lend people his house, his car, his wife, etc, only minor things).
If altruism is any good, people can want to do it voluntarily. Forcing someone else to be altruistic or self-sacrifice is only what people do when they don't know convincing reasons that it's best to live that way. It's also pretty mean to make someone else do something he doesn't want to. Surely being mean to them isn't a good way to teach children to care about other people.