Can we interpret sets of the same cardinality as distinct representations of the same set? Usually mathematicians consider isomorphic fields as equal fields. That is, if the $(A,+,\cdot)$ is isomorphic to $(B,\oplus,\odot)$, then I can consider those fields as equals. Thinking about it, I thought about the following interpretation:
Let $A$ and $B$ be two sets. I think we can interpret that $A$ and $B$, considered only as sets, are the same set if there is a bijection $\varphi:A\to B$ between them, because, just as the apparently distinct sets $\{1,\, 2,\, 3,\, \cdots\}$ and $\{I,\, II,\, III,\, \cdots\}$ can represent the set of natural numbers, $A$ and $B$, if there is a bijection between them, can be understood as distinct representations of the same set.
One can argue that this interpretation is wrong since there is a bijection between $\mathbb{N}$ and $\mathbb{Q}$ and these sets are clearly distinct since the first one has the well-ordering principle the second one doesn't. However $\mathbb{N}$ and $\mathbb{Q}$ are not simply sets, but an algebraic structures. For example, $\mathbb{N}$ is actually a triple $(\mathbb{N},+,\cdot)$ in which $\mathbb{N}=\{1,\, 2,\,3,\, \cdots\}$ and $+$ and $\cdot$ are binary operations defined in $\mathbb{N}$ with which we can build a well-ordering $\leq $ order in $\mathbb{N}$. So, as algebraic structures, $\mathbb{N}$ and $\mathbb{Q}$ are indeed distinct, but as simply sets we can consider them the same, because since there is an bijection $f:\mathbb{N}\to\mathbb{Q}$ I can represent all elements of $\mathbb{Q}$ with the symbols $\{1,\, 2,\, 3,\, \cdots\}$. To do this, I can simply represent an element $q\in\mathbb{Q}$ by an $n\in \{1,\, 2,\, 3,\, \cdots\}$ that satisfies $f(n)=q$. 
If we think well it is not possible to infer that $\mathbb{N}=\{1,\, 2,\,3,\, \cdots\}$ from the axioms of Peano. What happens is that we use the symbols $\{1,\, 2,\, 3,\, \cdots\}$ to represent the natural numbers which is the same attitude as representing $q\in\mathbb{Q}$ by an $n\in \{1,\, 2,\, 3,\, \cdots\}$ satisfying $f(n)=q$. Therefore, if we consider $\mathbb{N}$ and $\mathbb{Q}$ only as sets (and not as algebraic structures), we can say that $\mathbb{N}=\mathbb{Q}$ since it is possible to represent $\mathbb{Q}$ with the same symbols used to represent the elements of $\mathbb{N}$.
My question is: can we interpret sets of the same cardinality as distinct representations of the same set? Because if the answer is affirmative then it becomes easier to understand why we can consider isomorphic fields or isomorphic groups as equals. Thinking about groups I think it is also possible to think that if the groups $(G,\cdot)$ and $(L,\odot)$ are isomorphic, then $(G,\cdot)$ and $(L,\odot)$ are distinct representations of the same group.
EDIT: I think I've found a way to express myself better. I can say that $\mathbb{Q}=\{1,2,3,\cdots\} $ in which the element $5$, for instance, is understood as the rational $q$ satisfying $q=f(5)$ ($f:\mathbb{N}\to\mathbb{Q}$ is a bijection). Therefore, treating $\mathbb{Q}$ and $\mathbb{N}$ only as sets (and not as algebraic structures) I believe it makes sense to say $\mathbb{Q}=\mathbb{N}=\{1,2,3,\cdots\}$.
 A: Two sets are considered equal iff every element of the first set is an element of the second set and every element of the second set is an element of the first set.
Sets of the same Cardinality are not the same sets but they enjoy having a one to one correspondence between them.
You have answered your question when comparing the two sets , natural numbers and rational numbers.
They are not the same set. 
A: I suppose you know the concept of a category. If not, I strongly recommend to consult any book on this thematic area. Your question suggests that you regard objects of a category $\mathcal{C}$ as essentially equal if they are isomorphic in $\mathcal{C}$.
The category $Set$ has as objects all sets and as morphisms all functions between sets. The isomorphisms in $Set$ are precisely the bijections. This means that two sets are isomorphic in $Set$ if and only if they have the same cardinality. In that sense they are essentially equal. Your example $\mathbb{N}$ and $\mathbb{Q}$ does not concern the category $Set$, but the category $OrdSet$ of ordered sets (morphisms are order preserving functions). In this category $\mathbb{N}$ and $\mathbb{Q}$ with their natural orders are not isomorphic.
A: The thing to recognize here is that the mathematical notion of equality has bifuricated. 
In addition to the traditional notion of equality, we now recognize the value in considering the distinct notion of having an equivalence between two things.
We might also consider the proposition asserting two things are equivalent (i.e. that there exists an equivalence between them), but in my opinion practice has shown that's not the right notion; it's merely a frequently useful simplification. 
For sets, the right notion of equivalence is a bijective function. More generally for algebraic structures (or objects of a category) the right notion is that of an isomorphism.
If you only ever ask questions about equivalence, never about equality, then you will indeed see $\mathbb{N}$ and $\mathbb{Q}$ as being "the same" — and, in fact, see them as being "the same" in lots of different ways — because what you mean by "$\mathbb{N}$ is the same as $\mathbb{Q}$" is a bijection $\mathbb{N} \to \mathbb{Q}$.
Furthermore, you do see mathematicians in such settings repurpose the term $=$ to mean equivalence, and maybe even pronounce it as "equals" as well.
However, if you find yourself needing to also consider the traditional notion of equality — or, at least, want to converse with mathematicians who use the traditional notion — then you should use "equivalence" for the notion of sameness that you have.
