[in the article] there seems to be a confusion between dependency and functional dependency.
The article is using "dependency" in the sense of 'Dependence Logic' here or here. As @Mark R points out. Specifically it's talking about the 'branching or Henkin quantifier'. Yes those are nothing to do with database theory Functional Dependencies. (They might be a little to do with other forms of database Dependencies, see below.)
The expression $∀x∃y:S(x,y).$ ...
is not logically equivalent to $∃y∀x:S(x,y).$ (quantifiers permuted).
The first says (to stick with Lipton's example) 'every number has at least one square root' (True). The second says 'there's a number which is a square root of every number' (False).
But note that for the first formula to be true, requires you pick different numbers for $y$, for each different $x$. That's the sense in which $y$ is dependent.
The predicate $x^2 = a \wedge y^2 = b$ that Lipton uses as an example contains two functional dependencies.
You and I can see that (with our data analyst hats on). Lipton is ignoring it and looking at the (logical) dependency the other way round: having chosen $x$, there's always at least one choice for $a$. Database Dependency Theory has a name for that: it's a MultiValued Dependency.
Futhermore what he's really concentrating on is that $a$ depends on $x$ but not at all on $y$; $b$ depends on $y$ but not at all on $x$. Database Dependency Theory has a name for that, too: Join Dependency. A MultiValued Dependency is a special case. We can write this case as
$$
(JD) S: \bowtie(\{x, a\}, \{y, b\})
$$
That is, the relation representing predicate S can be vertically partitioned into two by projecting on those two pairs of attributes; and when we join them back together ($\bowtie$ symbol), we must get back the original relation value. There's a FOL equivalent for that, similar to your formula for an FD:
$$\begin{align}
\forall x, a, y, b, x', a', y', b' & [S(x, a, y, b) \wedge S(x', a', y', b')\\ & \Longrightarrow S(x, a, y', b') \wedge S(x', a', y, b)]
\end{align}$$
It seems to me we can use that to give the semantics of Lipton's example. without needing to escape into Second-Order logic. OTOH I think this is such a poorly chosen example, that might not be telling us much about the Branching/Henkin quantifier in general.