The letter $\upsilon$ or $\Upsilon$ is the second letter of δύναμις, κύβος and δυναμοδύναμις (square, cube and bisquare) of the words the notation abbreviates. Being the same letter, it plays an additional role of identifying the power symbols as being of a kind, marking Diophantus's "numerical species" (squares, cubes, etc., multiplied by coefficients). Notably, his power symbols have the same exponents, whereas ours have the same base. In other words, he thinks of them not as we do, a power marker modifying the "unknown", but in reverse, the implicit "unknown" modifying a generic power, given by the base of the power symbols. As Hettle puts it in The Symbolic and Mathematical Influence of Diophantus’s
Arithmetica: "Diophantus defined powers of unknowns not as a power of the unknown, but as an unknown with the property of being that power of a positive rational number; accordingly he refers to each power as a “species” of number".
In some sense then the letter $\upsilon$ is the mark of genericity, or of the unknown. However, the unknown by itself is denoted ς, a contraction of the first two letters of ἀριθμος, "number", according to Heath, which Diophantus uses to name it in the text. In contrast to the earlier tradition going back to Pythagoreans, he for the first time admits (positive) rationals as "numbers", and they are the only possible values of the unknown and legitimate solutions to equations he considers. Although it would seem natural to us to assimilate ς as marking "linear species", the difference in notation suggests that he does not think that way. Perhaps, this is a trace of Pythagoreans not viewing $1$ as a "number".
Numerical species are also sharply separated by Diophantus, notationally and terminologically, from free terms, to us, "zero power species", marked by $\mathring{M}$ for monad. They are Pythagorean and Euclid's "numbers" (positive integers greater than $1$, i.e. multitudes of monads) to which square, cube, etc., can be applied numerically. Apparently, he was concerned with possible confusion of his species with perfect squares, cubes, etc., even aside from the fact that zero concept, let alone zero power, was not available at the time.
Here is a more detailed commentary from Acerbi's Unaccountable Numbers:
"At the beginning of his treatise, Diophantus explains the
notation that he will use throughout; he is the first Greek
mathematician who consistently adopts a set of signs in order
to make his text more concise and, in a sense, conducive to the
kind of “algebraic” manipulations forming the technical core of
his method for solving numerical problems. In particular, he
establishes a terminology to denote what in algebraic language are the powers of the “unknown” $x^2, x^3, …$; in Diophantus’
theoretical framework, these are abstract numerical εἴδη
“species,” namely generic square, cube, … numbers. The
species introduced are assigned a denomination and a conventional
sign; the sign is made of the first letter of each component
of the denomination, possibly supplemented with the
second letter (this always happens to be upsilon): to the generic
square number (δύναμις) corresponds the sign $Δ^\Upsilon$, to the κύβος
the sign $K^\Upsilon$, to the fourth power (δυναμοδύναμις) the sign $Δ^\UpsilonΔ$, etc... At the end of the list of species, Diophantus also assigns a
denomination and a conventional sign to the most generic abstract
number, namely one that neither is a particular number
nor can be said to have the features characterizing one of the
aforementioned species; I shall call it, with a slight abuse of
language, the “1-species”; it corresponds to the “unknown” of present-day algebra.
[...] Capital $Δ$, $K$, and $\Upsilon$ are currently printed, but
of course no indication to that effect is contained in the text. It is quite
obvious that our notation owes very much, both in conception and in the
form of the signs, to Diophantus’: note his use of the term δύναμις “power”
and the idea of putting a part of the conventional sign “at the exponent.”
One crucial difference is that we conceive of the species as powers of the
“unknown,” whereas Diophantus draws a sharp distinction between these
notions, as we shall see presently. This difference is made particularly conspicuous
by the fact that Diophantus’ conventional signs all have the same
exponent (the insignificant letter upsilon) and a variable “base” indicating the
species (letters $Δ$ and $K$, possibly doubled), whereas modern algebraic signs
all have the same base (the most significant “unknown” $x$) and a variable exponent
indicating the power to which the base is to be raised.
[...] These species must not be confused with particular numbers that happen to be square, cube, fourth powers. On a terminological level, Diophantus settles the problem of separating
particular square numbers from the species “square” by
means of the opposition τετράγωνος/δύναμις; a lexical ambiguity
(admittedly quite harmless) remains in the case of the
κύβος, which may designate both a particular cube number
(such as $8$) and the species “cube.”
Diophantus highlights this difference when he alludes to the Euclidean
definition of number (Elem. 7.def.2) and when he defines a square number:
in both cases he adds a τινος, either to πλήθους or to ἀριθμοῦ. This means
that the object so qualified is particular, yet generic... The following considerations
may help further clarify the point. Diophantine numerical
species were invoked by the fourth-century mathematical polymath and
commentator Theon of Alexandria (In Alm. 452.21–453.16 Rome) to explain
the structure of orders within the sexagesimal system used by the
astronomers. The sexagesimal orders are in fact numerical εἴδη; they correspond
to the orders of magnitude in the decimal system... these numerical species do not coincide with particular numbers: indeed, $300$ is not a square, but $3$ items of the square εἶδος “hundreds”;
conversely, the species “hundreds” is not a number (it does not even
coincide with number $100$)."
