According to Wikipedia,
An integer sequence is a definable sequence, if there exists some statement P(x) which is true for that integer sequence x and false for all other integer sequences.
Surely, for any sequence of integers $x$, the statement $$ P_x(y):=y \text{ equals } x $$ would make $x$ a definable sequence. What am I misunderstanding?
The Wikipedia article is speaking very loosely there. The author doesn't say what language $P(x)$ is a formula or sentence of. If it can be any language, including all possible languages, including natural language, then you're right, everything becomes, trivially, "definable": for any given entity, there's always a predicate that's true of that thing and that thing only.
As used in mathematics, definability is always relative to a logical theory. Some integer sequences are definable in first-order Peano arithmetic; others are definable in second-order but not first-order Peano arithmetic; ZFC defines more still.
Though you don't bring it up, it's clear that the author has to mean infinite sequences of integers: every finite sequence of integers is definable in very minimal systems of arithmetic. In fact, for any finite structure you can come up with, there's a corresponding finite formula (in the appropriate language to represent your construction) which uniquely characterizes it. (It's very much like first order logic sentences which, if true in a model, forces the model to have at least 5 elements, or at most 4, or exactly 17; the bigger the numbers and structures, the bigger the formulas.)