A few preliminary responses to the OP's questions: Yes, a whole tone can be divided into any number of parts. Traditionally in the modern Western scale, we have 12 semitones per octave, where each whole tone is divided into 2 semitones. But there is no reason we can't divide musical intervals in other ways, including dividing a whole tone into 5 parts (or 3 or 4 or 10 or 200). In fact, we have a system for dividing every semitone into 100 parts, giving the unit called the cent.
And yes, a whole tone can be divided equally into two parts, but only if you use irrational numbers. Medieval music theorists, following in the tradition of Pythagoras, did not believe it was valid to tune musical intervals to irrational numbers and ratios. To tune the modern equal-tempered whole tone, you need to use a ratio involving 21/6. The modern equal-tempered semitone involves 21/12. These irrational numbers were literally viewed as irrational (i.e., defying reason) in the medieval era when it came to musical tuning. And dividing the vast majority of musical intervals precisely into two equal parts involves the square root of two, which was simply not done at this time. Hence, Marchetto "proves" that one can't divide the whole tone into two equal parts, just as would be true of most basic musical intervals for Pythagoreans.
Anyhow, on to Marchetto: The answers so far have speculated on Marchetto's tuning, but none of them seem to actually address Jan Herlinger's interpretation. Yes, there are lots of debates and conflicting interpretations among modern music theorists about what Marchetto actually meant by his talk of fives and nines, but the OP's question specifically pointed to an article by Herlinger.
And Herlinger clearly thinks Marchetto was dividing the Pythagorean whole tone (8:9 ratio, about 204 cents) into roughly five equal parts, that is, roughly 41 cents for each part. But Herlinger admits that this may have been meant as an approximation or sort of rule of thumb, as Marchetto elsewhere identifies 2/5 of a tone with the Pythagorean limma or minor semitone (243:256 ratio, about 90 cents) and his 3/5 of a tone with the Pythagorean apotome or major semitone (2048:2187, about 114 cents). And in yet another part of the treatise, Marchetto says the diatonic semitone or apotome can be viewed as a 16:17 ratio (about 105 cents), while his enharmonic semitone or limma was a 17:18 ratio (about 99 cents). Clearly there's some inconsistency in his implications. If you strictly follow a division of a 8:9 whole tone into five equal parts, they would result in unequal intervals of about 82 and 124 cents. Not quite like the traditional Pythagorean semitones, but in the same ballpark.
Herlinger goes further to suggest that the 1/5 and 4/5 tone intervals were not meant to be interpreted literally, but rather were approximations of the practice used then (and now) where, for example, leading tones are raised a bit near cadences to create tension to resolve. That would make a lot more sense from a practical musical standpoint than a literal equal division of the 8:9 whole tone into five parts. Herlinger elsewhere makes this explicit in his article "Fractional Divisions of the Whole Tone" (p. 78):
If [Marchetto's] words are taken literally, the leading tones will be
much too high: [In his examples] the major sixth would measure 955
cents, the major third 455 cents, intervals closer to a minor seventh
and a perfect fourth respectively. But since Marchetto intended the
measurements of the two-fifths and three-fifths tone as
approximations, could he not also have intended those of one-fifth and
four-fifths tone as approximations? If so, he would have been
described a system that showed the two traditional Pythagorean
semitones as well as another pair just a bit more different in size,
associated with progressions from imperfect consonances to perfect
ones.
Herlinger goes on to note that such an intonation system would be particularly appropriate for the sonorities in musical compositions of the Trecento (14th-century Italian music), which used the types of chromaticism in Marchetto's examples.
Whether or not we want to view this interpretation as definitive, Herlinger wrote his entire dissertation on Marchetto, published this (and other articles) and then published an annotated translation of the Marchetto treatise. He considered Marchetto's ideas in historical context, so we might grant some weight to Herlinger's interpretation, as he has likely spent more time with this work than any other modern scholar.
It's true that viewing the 1/5 tones as somewhat flexible can result in a rough approximation to 31-tone equal temperament. And it's also true that some people have focused on the role of 9 in Marchetto's division, assuming that Marchetto intended to divide the whole tone into 9 parts (as described in a link in another answer). And here's yet a different interpretation of the 9-fold division idea. (Herlinger addresses why the Latin text probably isn't intending a 9-fold division in footnote 20.)
Suffice it to say that Marchetto's exact intention for tuning is unclear. What is clear, however, is that actual medieval and renaissance music theorists seemed to assume that Marchetto was proposing primarily a 5-fold division, not a 9-fold one. Herlinger goes into detail in both the article linked by OP in the question and in my linked Herlinger article about how Marchetto was received. Johannes Tinctoris, perhaps the most well-known music theorist of the fifteenth century, echoed Marchetto's five-fold division in his brief definitions of intervals, for example, identifying the major semitone or apotome with 3/5 tone and the minor semitone or limma with 2/5 tone.
Clearly, music theorists of the time were trying to come up with some sort of approximation to these two unequal Pythagorean semitones. Herlinger cites Goscalcus Francigena (a treatise from c. 1375) with a 3-fold division of the whole tone, also intended to approximate the Pythagorean limma and apotome. Yet elsewhere this same author gives ratios that clearly show the limma was the standard 243:256 ratio. Like Marchetto, this treatise was probably treating 2/3 and 1/3 as approximations, not necessarily to be taken literally.
For some further context, Herlinger also cites the later theorist Gaffurio (c. 1480), who clearly intends a 9-fold division of the whole tone. Gaffurio discusses a division of the whole tone into four dieses and one comma, where the comma is half of a diesis. A diesis is a generic word in tuning theory for a small interval and can be vague, but a comma in this context is pretty clear: it's referring to either the Pythagorean comma of ratio 524288:531441 (about 23.5 cents) or the syntonic comma of ratio 80:81 (about 21.5 cents), either of which would make perfect sense as about 1/9 of a whole tone.
And that's yet another clue that Herlinger is probably on the right track in his interpretation, and the links in other answers hinting at a 9-fold division are off. In Marchetto's time, there was a clearly defined name already for the ratio of about 1/9th of a whole tone: it was called a comma in a number of 13th and 14th century treatises. The linked interpretation (by Joe Monzo) tries to interpret one of Marchetto's "fifths" as a 80:81 comma, making the other four roughly twice the size (dieses?). There's simply no evidence in Marchetto that he was intending the tuning discussed by Gaffurio a century later. If he were, he likely would have used the term comma in the appropriate place, or at a minimum differentiated the relative sizes of the dieses somehow. (The other historical implication of the word comma in tuning is "the small part left over" after breaking down some larger interval. It would be the precise word Marchetto would likely have used if he meant to divide the whole tone into four "big parts" and one "smaller leftover bit," i.e., the comma.)
Given Marchetto's lack of detailed tuning instructions for his "1/5 tones" and general imprecision compared to other treatises about such a novel tuning idea, it's reasonable to assume his dieses are referring to the ancient Greek implication of that word to mean "very small interval" without necessarily a specific ratio. That said, there is a very confusing passage discussing 9, but Herlinger explains that as simply referring to the 9 in the 8:9 ratio of the whole tone, which would require a 9-fold division of the monochord to tune properly. Maybe that 9 could be interpreted differently, but I've tried to outline Herlinger's reading of Marchetto that the question asked about.
Oh, lastly, Herlinger does give an explanation of why the number nine is so important. See, for example, footnote 17: "In associating the number 9 with perfection, Marchetto stands in the mainstream of Pythagorean numerology." Nine was frequently used in Pythagorean numerology to represent a generic idea of "totality." In subsequent pages, Herlinger gives several source quotations on why nine and odd numbers in general are important to Pythagorean music theory and the whole tone.
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