In short, anyone can.
It might seem odd but superannuation law is very explicit about who the directors should be (in most cases, the same as the members) but completely silent on who must own the shares in the company.
I say “odd” because of course shareholders of companies often have a lot of power. If nothing else, they usually choose the directors! But despite all that, shares in SMSF trustee companies can be owned by anyone without failing the SMSF rules set out in legislation. (It’s worth bearing in mind that sometimes the company’s constitution – it’s rule book – imposes its own rules about shareholders, and might even restrict shareholders to being the members of the SMSF. Those rules definitely need to be followed or the constitution should be changed, they can’t just be ignored. But they are not rules imposed by the law.)
In practice, most SMSFs have just one member or two members of a couple. The most common shareholding structure is that these same people are the shareholders. But different structures can be useful sometimes.
For example, what about an SMSF where the members are no longer directors of the trustee company? This might happen because they have moved overseas and asked someone else (usually someone who holds an enduring power of attorney) to be the director of the trustee company instead. Usually, the members would keep the shares in the trustee company. It means they keep quite a bit of control. While the new directors must be left alone to make all the decisions about the SMSF, the members (as shareholders in the corporate trustee) could always replace them if things got nasty.
Or similarly what about an SMSF where mum and dad (the original members) invite their four children to also belong to the fund? Again, there would usually be nothing to prevent mum and dad continuing to own all the shares in the corporate trustee. If the children are adults, they would need to be directors (or have someone who holds an enduring power of attorney for them be a director in their place) but ownership of the shares in the trustee company would still give the original members just a little more power. In fact that might make perfect sense if the family expects that in the long term, the children will move to their own fund. Why bother complicating things with share transfers at that time? Why not just leave the shares in the hands of the parents throughout?
It's worth noting that even in this case the parents (original members) might find that their control over the fund is not as significant as they thought. For example, SMSF trust deeds often allow the members to replace the trustee entirely. If that was the case, a majority of members might be able to simply remove the original trustee company (owned by mum and dad) and replace it with a company in which the shareholding was entirely different.
So overall, even though super law doesn’t require it, the most common approach is for the directors, members and shareholders all to be the same people. Cases where some of these are different usually come about to achieve a specific outcome in relation to control or estate planning.