Does "=" have to be interpreted as equality?
To put it briefly: In model theory, we are allowed to interpret any relation symbol in any way we like. So why do people seem to require that "$=$" is interpreted as the actual equality?
Let me elaborate a little more. In model theory, as I imperfectly understand it, one starts with an alphabet $\Sigma$ consisting of the allowable function and relation symbols; for instance for ordered fields we could take $\Sigma = \{\cdot,+,<,0,1\}$. To these, we add symbols for variables $x_1,x_2,\dots$ and logical symbols $\vee, \wedge, \forall, \exists, \dots$. Using these, we can form terms (well formed expressions that will describe elements of the set) and sentences (expressions that will either be true or false). If we assume some set $A$ of sentences to be true (axioms), then the set of all their logical consequences, say $T$, is a theory. A theory can be interpreted by first choosing a set $X$ to work on, and then assigning to the function and relation symbols actual functions ($X^k \to X$) and relations ($X^k \to \{\top,\bot\}$). This is to be done in such a way that the axioms are satisfied.
My problem is that the equality seems to be treated in a different way than other relations, and I don't quite see why. As far as I understand, it is normally required to be the "real" identity: $x = y$ means that $x$ and $y$ are the same element of $X$. Is there some reason not to treat "$=$" just as an ordinary relation (with axioms of being an equivalence relation + for each relation symbol axiom "if $x_1 = y_1,\dots,x_k=y_k$, then $R(x_1,\dots,x_k)$ iff $R(y_1,\dots,y_k)$" )? What would go wrong if we did?
(The reason I am asking is mostly that it seems to me that this would make some constructions more elegant (such as the ultraproducts) by eliminating a quotient.)
It is not wrong. We can interpret '$=$' as an arbitrary binary relation symbol, then if we insert the statements to $A$ that it is equivalence relation and that it preserves all other relation and function symbols, then it will be interpreted as a congruence relation $\sim$ on a model $X$, and we can calmly form the quotient model $X/\sim$, which behaves exactly the same way as $X$, and '$=$' will be interpreted as real equality therein.
To add a historical twist to the fine answers given earlier, note that Fermat, Leibniz, and Euler precisely considered a generalized notion of equality in their work. Fermat is discussed here: http://www.mitpressjournals.org/doi/abs/10.1162/POSC_a_00101, Leibniz here: http://www.ams.org/notices/201211/rtx121101550p.pdf, and Euler here: http://u.cs.biu.ac.il/~katzmik/bairetal.html
When I learned logic in college, the equality relation was treated specially: rather than taking = as a possibly optional symbol in a particular language, it was a built-in feature that had the same syntactic role as any other relation symbol, but was required to have the interpretation of actual equality in a model. That is, it had the same status as $\neg$, which one could interpret as a unary relation (i.e. a predicate) but is in fact fixed in meaning by the rules of logic.
There is no disadvantage to having a symbol with a certain requisite meaning in every model, so long as that meaning is available regardless of the language. For one could otherwise just simulate it with a relation symbol, as you say, except that we'd get other (presumably spurious) models with = not meaning equality. Of course, since almost any set with any structures can be a model for some theory, there isn't much room for this kind of thing; however, equality is one relation you can guarantee exists, so we do.
No, there is nothing wrong with treating "$=$" as just another relation, although that approach seems to be out of fashion. See Abraham Robinson's 1965 book Introduction to Model Theory and to the Metamathematics of Algebra. If memory serves (I don't have a copy) it treats equality the way you are asking for.