I have been going through Velleman's How to prove book and in one of their sample problems they have used ∀x∀yM(x, y) for representing all married couples where
M(x,y): x and y are married to each to each other
But on deeper thought, I'm confused by this statement. Let the Universe of discourse be all peoples and let us substitute x as Joey. Now the statement will be like this:
∀yM(Joey, y)
which roughly translates to: for all people y, Joey is married to y.
But isn't that false because a person can't be married to everyone.
UPDATE: How actually Velleman uses it:
"All married couples have fights" is translated to
∀x∀y(M(x, y) → F(x, y)), where M(x, y) means “x and y are married to
each other” and F(x, y) means “x and y fight with each other.”
If Velleman says what you claim he did, he is making a mistake. Or more likely, you misunderstood.
Expressions like $$\forall x. \exists y . \text{$y$ is the mother of $x$}$$ are statements; they are either true or false. (This one happens to be true.) They do not represent sets of any sort.
However, one can use the so-called set-comprehension notation to represent the set of things of which some statement is true. The notation $\{x \mid P(x)\}$ represents the set of all things $x$ for which the statement $P(x)$ is true. For example:
$$\{x \mid \text{Anne is the mother of $x$} \}$$ is the set that contains every person of whom Anne is the mother: it is the set of Anne's children, in other words.
Your example might be $$\{\langle x,y\rangle \mid \text{$x$ and $y$ are married}\}$$ which is the set of all pairs $\langle x, y\rangle$ where $x$ and $y$ are married. One of the many pairs in this set is the pair $\langle \text{Queen Elizabeth II}, \text{Prince Philip}\rangle$. If Velleman claims that this set is represented by the expression $$\forall x\forall y. \text{$x$ and $y$ are married}$$ then he is wrong; if he said something else that you understood to mean this, then you misunderstood. As you observed, $\forall x\forall y. \text{$x$ and $y$ are married}$ is a statement, not a set, and it happens to be false, since it asserts, among other things, that Michael Jackson is married to Julius Caesar.
Similarly, $$\forall y.\text{Joey is married to $y$}$$ asserts that Joey is married to everyone, which is false. But $$\{y\mid \text{Joey is married to $y$}\}$$ represents the set of all people that are married to Joey.
You have updated the question to make clear that it concerns the statement $$\forall x\forall y . \text{$x$ is married to $y$} \to \text{$x$ fights with $y$}.\tag{$\star$}$$ This is a statement, intended to assert that if $x$ and $y$ are married, then they fight with one another, or colloquially “all married couples fight".
It does not assert that everyone is married to everyone else, which would be $$\forall x \forall y . \text{$x$ is married to $y$}.$$ It only asserts that for any $x$ and $y$, if they are married then they fight.
Let's suppose this is actualy true, that all married couples do fight. As you observe, we should be able to put in anything we want for $x$ and $y$ and obtain a true statement. Let's take $x=\text{Michael Jackson}$ and $y=\text{Julius Caesar}$. Then the statement asserts, for this example, that
$$\text{Michael Jackson is married to Julius Caesar} \to \text{Michael Jackson fights with Julius Caesar}.$$
That is, if Michael Jackson and Julius Caesar are married, then they fight. This is an implication, and since the antecedent is false, the entire statement is true whether or not they fight. So even though $x$ and $y$ are not married in this case, the statement is true anyway. (See the discussion of implication on page 43.)