The fact that you mention Replacement leads me to believe that you're especially interested in first-order theories which are meant to capture some sort of set theory. In a set theory, we have a way of identifying ordered pairs with elements of the universe - the standard choice in ZFC is to code $(a,b)$ by the element $\{\{a\},\{a,b\}\}$.
Now we have a definable predicate Pair($x$) which determines whether some set $x$ codes a pair (it says "This set has two elements. One of these has one element, $a$. The other has at most two elements, one of which is $a$.") and definable functions First($x$) and Second($x$) which return $a$ and $b$. The theory then proves $\forall a \forall b \exists x\, \text{Pair}(x) \land (\text{First}(x) = a) \land (\text{Second}(x) = b)$.
By adding the ordered pair function, you're adding a Skolem function for the sentence above. This eliminates the existential quantifier, so you now have $\forall a \forall b \text{Pair}((a,b)) \land (\text{First}((a,b)) = a) \land (\text{Second}((a,b)) = b)$. This is a totally standard thing to do in first-order logic - it simplifies syntax without changing the strength of the theory or its class of models (this is what the user aws means by saying that set theory has a "conservative" extension with a pairing operator). It's analogous to adding the inverse symbol ($^{-1}$) to the theory of groups. You can read more details on the wikipedia link.
But set theory is somehow special in that there is a canonical way to code a pair of elements as a single element. That is, there is a definable injection $M^2 \rightarrow M$ for any model $M$. In general first-order logic, you don't have this kind of structure, and adding a pairing function to a general theory could really change the theory.
However, there is a "safe" way to add pairing (and n-tuple) functions to an arbitrary first-order theory, which I'll describe now. We move from single-sorted logic to many-sorted logic, with one sort for each natural number. The $n^{th}$ sort represents the cartesian power $M^n$. Now for each $n$, we can also add an $n$-ary function $f_n$ which takes the $n$ elements $x_1,\dots,x_n$ to the element in the $n^{th}$ sort representing the $n$-tuple ($x_1,\dots,x_n$). We also add axioms stating that $f_n$ is a bijection for each $n$. One can check that this is construction gives a "conservative extension" in some sense.
The advantage of this situation is that we can now view all definable functions and relations as unary, as you wanted. The expense is that we're in a many-sorted context, which you have to get used to. Model theorists often go further, adding new sorts not just for all $M^n$, but for all quotients of $M^n$ by definable equivalence relations - the resulting theory is called $T^{eq}$, and it is a nice place to work for a variety of reasons.
