You've stated that you want to ignore any instability involved, so I'll skip that. There are three eventualities I can think of that might work, one depending on another potential loophoole.
Generally speaking, it's a capitalist system, so some people are going to end up with more shares than others. The greater number of people in important positions you control, the more power you have.
1) Uneasy Truce
A lucky investment, the father of a child prodigy; like with companies there are a number of ways that someone could strike it rich. Given that in this case you literally own the % assets of the person related to your shares in them, once you make it rich it's rather easy to stay rich... provided you don't get bought out. That's where the "Government" comes in.
A number of people have become very rich, and they plan on staying that way. They know that if they only work for themselves, the chances are other people will gang up on them and forcibly remove their assets, so they make an agreement: I don't touch yours, you don't touch mine, and together we "own" enough people performing enough important services to dictate what happens in this society. They could even shuffle their assets so that each effectively becomes a "minister" of the different governmental areas: One person owns a controlling stake in the high-authority positions in Health, another in Finance, another in Foreign Policy, say. Different ministries would be more lucrative, obviously, and that would define the relative levels of power within this group.
The group itself has no shareholders outside of their closed circle: through a combination of shrewd business deals and intimidation, they bought out everyone else before they realised what was going on, or by exercising control through other channels. Now, all they need to do is stop anyone else from getting as much power, and trust that each member is going to keep the truce.
Obviously this kind of system isn't going to stay stable for long, but it doesn't have to stay stable for long if you're just interested in watching it fall, and it's amazing how crazy a system people can rationalise themselves into as a society, and how long they'll stay there before enough people complain about it. It's entirely possible that such a system could form and last for at least 20, 30 years, if not longer.
2) The Co-operatives
As the dust settles after whatever ridiculous event caused this structure to form, people quickly realised that they couldn't make it on their own, but they had to be careful who controlled the people controlling them. Therefore "cooperatives" formed between people of similar mindsets and morals: by buying each other out and creating a closed circuit they reduced the risk of being forced to do something they didn't want to do. These cooperatives became the fundamental social unit in the society, with each cooperative electing representatives to take part in a joint council. Some are clearly more powerful than others, but the threat of being ganged up on by the combined buying power of the rest forced everyone into line... after a few examples had been made, of course. New cooperatives formed and disbanded over time as people tried to go in another direction, but they rarely managed to survive in the face of the established powers.
There would be power struggles within the cooperatives themselves, but they would be limited — regardless of how frustrated you might be with your current lot, it would be a lot worse if your cooperative became infiltrated by an outside power!
The key bit of information for me is what happens when someone dies. Where do their shares go? Are they passed on in a form of will? Do they go to next of kin? What happens to their children's shares? Under our inheritance laws they would "inherit" their own shares, which is technically illegal under your rules. Could I become a free man with all my assets if I killed all my shareholders? Presumably I have to agree before someone can buy shares in me (unless they're buying them from someone else), so if I have no shareholders but lots of assets due to all my shareholders dying, that leaves me in a position of great power.
This leads to:
3) The Private Limited People
"PLPs", as I will refer to them, are people that, by some method or another, have managed to amass a lot of shares in other people but have no shareholders in themselves. For most people, when they lose their shareholders, they don't have enough capital to keep themselves going without investment, so they have to go and seek people to buy more shares in them. Some people, however, have built a large enough portfolio that they can afford to effectively own themselves. It's not illegal because they haven't "bought" shares in themselves, they just haven't sold any to anyone living. These people become the most powerful people in the country, because they are accountable to no one. Everyone else can be influenced by some share trail or another, but these guys are totally untouchable. Plus, there are rumours about how they came to be shareholder-less, the mysteries that surround the sudden and tragic deaths of their parents...
Seriously though, I would think long and hard about death and what happens to shares in that situation. It can either be a key part of the plot that is integral in how the system becomes unhinged (in the uneasy truce, one of the main shareholders mysteriously dies and the balance of power shifts within the group shifts dramatically as suddenly the internal share structure isn't so balanced as it used to be), or it can be a massive flaw.
Any of the 3 above would be viable, at least on face value. There are likely flaws that would show up if you dug hard enough into the details. A combination of them might also work - start with 1 or 2, 3 happens and that's what changes everything.
