The value $\operatorname{sign}(x)$ equals $-1$, $0$, or $1$ depending upon whether the value $x$ is negative, zero, or positive. Sometimes this is written as $\operatorname{sgn}(x)$ as is the case here.
When we say $(A,\square,\triangle)$ is a field, we are saying that the base set of the field is $A$, that $\square$ and $\triangle$ are binary operations on $A$ satisfying certain properties.
We use an ordered triple because we need to distinguish between the first operation and second operation, they don't satisfy the same axioms. In the case of that field, for example, we know that $\triangle$ distributes over $\square$, but no the other way around, or that the every element of $A$ has an inverse with respect to the $\square$ operation, and that's not true for $\triangle$. If we write "let $A$ a field with operations $\#$ and $\&$" it's not clear from the statement which operation corresponds to the "addition" of field and which to the "multiplication". Writing them in a triple get rid of the ambiguity.
Also, I think you have a misconception, because you ask "how do you know the underling relation between these objects in the quadruple "(in this case, triple). The point is that the notation "$(A,\square,\triangle)$" alone is incomplete. The correct notation would be "$(A,\square,\triangle)$ is a field".
From $(x,y)$ we obviously can't know if that denotes a group or a metric space (or a completely different thing). Examples of COMPLETE notations are is "$(x,y)$ is a group" or "the metric space $(x,y)$".
Best Answer
Very generally, you could think of $\cong$ as meaning "two things are 'essentially the same' but are not identically one", whereas $=$ means they are identically one. This is better understood and explained by looking at the cases where it's used:
In geometry, $\cong$ means congruence of figures, which means the figures have the same shape and size. (In advanced geometry, it means one is the image of the other under a mapping known as an "isometry", which provides a formal definition of what "same shape and size" means) Two congruent triangles look exactly the same, but they are not the same triangle, as then there would be but one triangle, not two. (One could also say they are formed from different points in the space.) Colloquially speaking, they are "copies" or "clones" of each other. A copy of something is not literally identically one and the same as the other, even though they are alike in every respect.
In abstract algebra, $\cong$ means isomorphism, which says the two objects are structurally the same. Intuitively, if we have, say, a pair of groups, and they are isomorphic, the "patterns" formed by the operations are the same, even if the elements making up the groups' base sets are not the same. (That is, if you could draw a table (not physically possible for infinite groups) of the group operations, they would have the same "pattern", just expressed with possibly different symbols) Formally, it means there is a bijective map between the two which respects the operations (in the sense given in the other posts here). Any abstract-algebraic properties of one hold for the other (that is, any properties that do not depend specifically on the particular characteristics of the elements of the base sets as objects in and of themselves). From the point of view of abstract algebra, it is this structure that is what matters, and the precise composition of the base sets does not, so from that point of view we would say they are "essentially the same", but they need not be identically the same, for the composition of the base sets may differ. If the base sets are the same, then we really do have only one mathematical object, and they are equal, $=$.
There is probably some philosophical points here -- the distinction certainly does have a philosophical flavor to it, namely regarding whether or not things which are indiscernible (i.e. $\cong$) are identical (i.e. $=$). In mathematics, it is useful to make a distinction between the two concepts: indiscernible (i.e. congruent) triangles may occupy different parts of space, for example, and isomorphic groups may need to be distinguished if, say, we are dealing with them as part of a larger problem or situation in which we need to deal with more than just their properties from an abstract-algebraic point of view, i.e. where the composition of their base sets is relevant to the problem as well. There might also be cases where there are multiple types of structure defined on the same base set, and two objects may be isomorphic with respect to one kind of structure, but not another.