Setting and provisional answer
As your quote is a note regarding a certain sentence, I will first quote this one for providing context:
He who has the right to vote in this legislation is called a citizen (citoyen,
i.e., citizen of a state, not of a town, bourgeois). The quality requisite to this,
apart from the natural one (of not being a child or a woman), is only that of
being one's own master (sui iuris), hence having some property (and any art,
craft, fine art, or science can be counted as property) that supports him -
that is, if he must acquire from others in order to live, he does so only by
alienating what is his [!here comes the note!] and not by giving others permission to make use of
his powers - and hence [the requisite quality is] that, in the strict sense of
the word, he serves no one other than the commonwealth. (On the Common Saying... 8:295)
Having both parts together now, we can analyze what this is all about: It is about requirements counting as a citizen. The main condition is according to him being sui iuris, i.e. independent or one's own master qua property.
The difference between a barber and a wigmaker therefore is that the barber indeed has property in sense of a craft just as the wigmaker, but he is not independent through it because he never really posesses it, he necessarily "alienates what is his"
Alienation vs. production of property
So what does it mean that he alienates his property, i.e. his craft? Well, Kant is quite explicit about it:
... the former [wood cutter] differs
from the latter [tailor], as a barber from a wigmaker (even if I have given him the hair for the wig)
and hence as a day laborer from an artist or craftsman, who makes a work that belongs to him
until he is paid for it). (emphasis mine)
The difference therefore lies in the point that the barber, even when executing his craft, does never own (the fruits of) his work, whereas the wigmaker does. A barber is dependent on hoping to get paid for what he did (= alienated from him - his labour power) without ever posessing actual property to trade, i.e. he does not own the product of his work.
He is in this sense a service provider (to translate it into a modern term) and closer to a day labourer - offering nothing more than his labour power - than an artist/craftsman. This is also coherent with the difference between a wood cutter and a tailor.
Keep in mind, though, that the last sentence of the note clearly states that this criterion is problematic, as it was hard to determine when someone really is one's own master, implying that there is some kind of a grey area. Ironically, this makes the criterion correct theoretically, but of no use in practice.
From today's perspective, this view must seem odd. But indeed, it is an improvement compared to the normal ancient view, like Hannah Arendt depicts it in The Human Condition (Arendt is very detailed and shows this point for e.g. Plato and Plutarch as well as Aristotle, her main focus; this is only a very short outtake!):
What prevented the polis from violating the private lives of its citizens and made it hold sacred the boundaries of surrounding each property was not respect for private property, but the fact that without owning a house a man could not participate in the affairs of the world because he had no location in it which was properly his own. (29-30, 2nd edn.)
Here, property is the central concept as well. But instead of narrowing it to owning a house (and don't having to work for sustaining one's life at all!) as a necessary requirement for being a citizen, Kant allows for much more people to achieve the status of a citizen, i.e. a zoon politicon. That is a huge improvement! This reference could also be reflected in his distinguishing of citoyen [state and in reference to Rousseau] and bourgeois [polis].
The same thought of property grounding state and citizenship is found in Locke's Second Treatise of Government, coincidentally containing one of the few early bits of philosophy about the concept of work as well. I actually think this may be the main adressee here, although the section is supposed to be against Hobbes by its title. The big difference is that Locke simply assumes you make something your property through work, not even considering other cases.
The argument for revoking materially dependent persons the right to vote by many authors (e.g. Locke, Kant himself, and, very influential for Kant, Achenwall/Wolff in Germany and Abbe Sieyès in the wake of the French Republic) is that they are dependent. In fact, allowing them to vote would, so the argument says, result in them voting to please their masters or the persons they depend on. By that, those who own more would have more weight in the electorate than those who are independent as well, but own less. Or, to put it differently: Some citizens would effectively have more than one vote. So, given you want equal vote, you either disqualify some as active citizens (Kant's solution) or fight against material dependency as a whole (Marx' solution), see e.g. Vorländer, K. (1926). Kant Und Marx Ein Beitrag Zur Philosophie des Sozialismus.
One should keep in mind that voting by secret ballot in independent election was not exactly standard back then. Maliks (see below) quotes Allen Rosen writing:
those who depend on others for their livelihood would either be too eager to please their masters or too susceptible to pressure, particularly in an open ballot system, for their votes to be truly their own (p.81)
There is also a formal reason: Kant uses the Hobbesian criterion from De Cive that "[the citizen] serves no one other than the commonwealth". As again Maliks writes:
The reason is that women and the financially dependent are incapable of voting impartially because they answer to masters other than the state.
So e.g. civil servants are in some sense "dependent", but they are nevertheless "their own masters", as they serve no one other than the state. (p.108)
This view must seem odd in modern understanding, but was the practice of the time.
I would therefore argue that it must seem odd considering La Grande Révolution had happened already, the idea of fundamental equality etc. But especially regarding the philosophy of work, it is a considerable advance compared to everything that had been there before. Additionally, the development from Kant over Hegel to Marx' Theory of Alienation may be seen as presaged.
Further Reading and scholar's answer
Of course, the inner tension between moral theory and accounts of citizenship is widely discussed (and already had been in his times). For a historical view, see Maliks, Reidar (2014): Kant's Politics in Context, Oxford UP, Ch.3 (pp. 80ff). For two different contemporary approaches to this problem see James, David (2016): Independence and Property in Kant's Rechtslehre , British
Journal for the History of Philosophy, 24:2, 302-322 and especially Storey, Ian (2012): Kant's Dilemma and the Double Life of Citizenship, Contemporary Readings in Law and Social Justice, 4:2, pp. 65–88.
Maliks writes, referring to the argument laid out earlier (p.81):
Most scholars agree that he [i.e. Kant] was convinced that the votes of the many who were economically dependent would distort the general will, given his doubts about their ability to make free and rational choices about public matters.
In this text, there are further sources to be found for that argument probably being what Kant had in mind there as well.
In German, the translation of sui iuris - selbständig - coincedentally can be understood both as independent and self-employed, which may be understood as the modern equivalent of what Kant describes here.