Why logical equivalence sentences aren't considered as propositions? In the book Discrete Mathematics and Its Applications (by Kenneth H. Rosen), he defines logical equivalence as such

"The compound propositions $p$ and $q$ are called logically equivalent if $p\leftrightarrow q$ is a tautology, The notation $p\equiv q$ denotes that $p$ and $q$ are logically equivalent"

Then he said that "$p \equiv q$" is not a compound proposition but rather is the statement that $p \leftrightarrow q$ is a tautology. By definition of a proposition or a statement (a declarative sentence that is either true or false, but not both), I feel like $p \equiv q$ can actually have a truth value. If $p$ isn't equivalent to $q$, then saying $p \equiv q$ would be false. So, what's making $p \equiv q$ not considered a (compound) proposition? I feel like $p \equiv q$ can be true or false. If it's a proposition, then does $p \equiv q$ also mean $(p \leftrightarrow q) \leftrightarrow T$, where $T$ is a tautology?
I'm aware that a quite similar question has been asked here but it doesn't really answer the first part of my question. Though from the answer there, it mentions something about metalanguage. Does that mean that I need to learn metalanguage to have my questions answered? Or am I just mistaken here? Thank you.

 A: If by "proposition" you just mean a sentence that can be evaluated as true or false, then logical equivalences are propositions.
The difference he's getting at is that "$A \leftrightarrow B$" is a sentence of the object language (propositional logic), while "$A \leftrightarrow B$ is a tautology." is a sentence of the metalanguage (English), that speaks about the logical formula. In his terminology, the term "proposition" may be reserved for sentences of the object language. Hence his accentuating the fact that "$\equiv$" is not a logical connective, but rather short-hand for the English expression "is logically equivalent to".
A: Here is an example of equivalence:
$$ (a \rightarrow b) \equiv (\lnot a \lor b), $$
where $a$ and $b$ are simple (not compound) propositions.
It's an equivalence because the compound proposition
$$ (a \rightarrow b) \leftrightarrow (\lnot a \lor b)  \tag1$$
is true for all four possible choices of the truth values of $a$ and $b$
where $a$ and $b$ independently can be either true or false,
that is, proposition $(1)$ is a tautology.
On the other hand, $a \rightarrow b$ and $a \lor b$ are not equivalent.
We can of course write
$$ (a \rightarrow b) \leftrightarrow (a \lor b),  \tag2$$
but this compound proposition won't be true for all possible values of $a$ and $b.$
For example, proposition $(2)$ is true when $a$ is false and $b$ is true,
but it is false when $a$ is true and $b$ is false.
We could write the compound proposition
$$ ((a \rightarrow b) \leftrightarrow (a \lor b)) \leftrightarrow T $$
where $T$ is a tautology, for example,
$$ ((a \rightarrow b) \leftrightarrow (a \lor b)) \leftrightarrow 
((a \rightarrow b) \leftrightarrow (\lnot a \lor b)). \tag3 $$
You could read proposition $(3)$ loosely as,
"Proposition $(2)$ is a tautology."
But what it really says is closer to,
"Proposition $(2)$ is true if and only if a tautology (specifically the tautology of proposition $(1)$) is true."
As it turns out, proposition $(3)$ is true whenever proposition $(2)$ is true, and false whenever proposition $(2)$ is false.
(Try checking it.)
What is missing from proposition $(3)$ is the qualification, "And this is true no matter which truth value (true or false) is assigned to each simple proposition independently of all other simple propositions."
That's what $p \equiv q$ gives you.
It says no matter what else we suppose, these two propositions are either both true or both false.

I think it's perfectly reasonable to object that $p \equiv q$ could be considered a proposition according to a definition that only requires a proposition to be a declarative sentence that is true or false.
That's a rather loose definition of a proposition;
we really ought to explain more concretely how simple propositions can be formed and how compound propositions can be formed.
It's also reasonable (given such a loose definition) to ask why
"$X$ is a tautology" is not a proposition when $X$ is a proposition.
The reasons for these apparent discrepancies (as I understand them)
is that propositional logic was never intended to be a universal language encompassing all possible true-or-false declarative sentences in English
(or if it was once hoped to be such a language, it was soon discovered that it cannot be).
Instead, we build up from very simple languages to more complex ones
(such as the metalanguages mentioned in comments).
