The parent did not mention anarchy. Abolishing the concept of citizenship is not exclusive to having a functioning government.
If you govern a population without a land, you are practically instantly at war with some other population because both populations will want their way of life as well as their property rights enforced around the place they call "home". For a current example, see the palestinian exile population, who have a government but no land of their own, thus leading to constant conflict with their host countries.
If you govern a land without a population, you are lacking any kind of compass and attachment to values. Land alone is a dead thing, and a government cannot just be recruited from land, it has to be people doing the governing. Basically, there is nothing to govern without a society.
Citizenship can have a number of definitions, but the loosest one is something like "currently inhabiting the land area of that corresponding government". You may change those definitions, introduce various classes of citizenships, modify the ways in which it can be obtained. But for the aforementioned reasons, any definition that doesn't involve something like "a citizen is strongly associated with a land area and comes from a corresponding populace (governed by a corresponding government)" is a weak and fragile definition that will not last the test of time and human interactions. Note that the government part is in parentheses, because actually governments are far more interchangeable than population and land area.
The article claims they have no territory, but what they mean is there is no territory capable of supporting a military + institution. I don't know this is a practical counterexample.
Nevertheless, it is interesting to note that clubs and private militaries, and independent autonomous intelligence agencies exist, even if they are on the extreme end of what could be considered a "government".
You can call it something besides "citizenship", but maybe we're talking about different terms, or you have a more technical definition implying more than simply membership under a governing party?
Either way, this is the basic definition of a citizen, so trying to divorce the two seems futile to me:
"A citizen is a member of a political community who enjoys the rights and assumes the duties of membership."
Being in the position of abolishing _birthright_ citizenship is a very different stance than abolishing the practice of citizenship outright. I think you are speaking of "second class citizenship" e.g. immigrants or temporary who do in fact benefit from and are expected to follow the laws of the land, but who do not get e.g. voting rights.
I.e. perhaps the view is the equalize all practical citizens under the law (which is possible but difficult to do wrt to land ownership and community security, especially).
Countries would still be governed by those who live within their borders, but you wouldn't be discriminated against based on your place of birth, e.g., you could move around freely.
Or, maybe more practically, greatly ease the requirements of immigrating and becoming a citizen of countries.