UPDATE: Thanks to @Ian, I re-read the original source more clearly and the exact claim is false. The 6.6 billion is too high. Almost all of the Queen's ownership of land is a purely technical one, in which she represents the governments of the country. However, if you accept the technicality of the Queen being the owner of government owned land, the Queen is still the largest landowner on Earth by a large margin, just not nearly the margin that article claims. See below for details.
In British commonwealth legal tradition, land owned by the state is called "Crown lands". It is nominally titled to 'the Crown' of that country, but that actually means the government. This is still the case, for example, in Canada, by far the largest of the Crown holdings, where about 89% of the land is crown land. Approximately 23% of Australia is Crown land.
Much crown land is used for government purposes. Some generates income from leasing it, but she doesn't actually receive this money personally. Legally speaking, the land is held by the government "in the name of the Crown", and it uses the income from the lands to actually operate the government.
For example (thanks to @Brian M. Hunt pointing this out): The Canadian Constitution (specifically, the Constitution Act of 1867) enumerates the rights given to the provinces and those kept by the federal government. One of the rights delegated to the provinces is:
Property and Civil Rights in the Province. (Section 92(13))
meaning that the provinces have the right to make laws regarding property rights.
Mostly, this ownership is a legal fiction meant to allow land ownership and transfers to continue to operate as it always has. It would require an act of the respective Parliaments to change the status quo, and no one has felt the need to do so in Canada.
The 6.6 billion acre number quoted in the article is too high, because it includes land that is not even technically owned by Queen Elizabeth II. For example, the article includes
Australia, the 7th largest country on earth with 1,900 million acres, the Papua New Guinea with 114 million acres, New Zealand with 66 million acres and the UK with 60 million acres.
The Queen does not own any of the land in Australia; though they still follow the Commonwealth land tenure laws, "The Crown" in Australian law is specifically defined as the governments of the state or territory, and not the person of the monarch, so none of that 1.9 billion acres belongs to The Queen. Similarly, most of the crown lands in the UK were ceded by George III in 1760, and are held in an independent public trust called The Crown Estate, and not owned by Queen Elizabeth II. In New Zealand, only a small portion of the Crown Lands are still held in the name of the monarch (about 8% or ~7 million acres). The situation is Papua New Guinea is complicated, but it appears that as much as 97% of the land is Customary Land, which is owned in the indigenous people of the country.
Overall, I would estimate that Her Majesty's purely technical holdings are probably closer to about 2.5 billion acres, most of which is Canada and Australia, but that still puts her 5 times higher than the next largest land owner. The actual amount of land she owns in any meaningful sense is much smaller.