Some confusion and incorrect statements in the answers already given, including the "accepted" answer. (E.g. obvious distinctions between necessity and sufficiency of different conditions for KKT are mixed up.)
There are different formulations of KKT theorem.
The KKT Theorems one might consider for your example are the following.
For notational simplicity, $(x,y)$ will be replaced by $x$.
For the constrained optimization problem
$$
\max_{g(x) \leq 0} f(x), \quad (*)
$$
consider the KKT conditions
\begin{align}
\mbox{(i) } \nabla f &= \lambda \nabla g, \; \mbox{for some } \lambda \geq 0 , \\
\mbox{(ii) } \lambda g &= 0 \mbox{ (complementary slackness) }.
\end{align}
Theorem (General KKT)
If $x$ is a local maximum of $(*)$ where $\nabla g(x) \neq 0$, then (i) and (ii) holds at $x$.
The condition $\nabla g(x) \neq 0$ is called a "constraint qualification" condition.
In other words, the KKT conditions (i) and (ii) are necessary conditions for local
maxima where constraint qualification holds. It says nothing about global maxima.
Even a local maxima where constraint qualification fails need not lie in the solution set of (i) and (ii).
For your example, constraint qualification fails at the global maxima $x^* = (1,1)$, $\nabla g(x^*) = 0$. So General KKT doesn't apply.
Theorem (KKT Under Concavity)
Suppose $f$ is concave and $g$ is convex. If (i) and (ii) holds at $x$, then $x$ is a global maximum of $(*)$.
In other words, when $f$ is concave and $g$ is convex, then (i) and (ii) are sufficient condition for global maximum.
In your example, $f$ is concave and $g$ is convex. But the solution set of (i) and (ii), with $g \leq 0$, is empty---these are vacuous conditions. So KKT Under Concavity doesn't apply.
Theorem (KKT Under Concavity and Slater's Condition)
Suppose $f$ is concave, $g$ is convex, and Slater's condition ($\{g <0 \}$ is non-empty) holds . If (i) and (ii) holds at $x$, then $x$ is a global maximum of $(*)$.
In other words, under Slater's condition (i) and (ii) are now necessary for global maximum.
In your example, Slater's condition doesn't hold. Indeed, the KKT conditions (i) and (ii) cannot be necessary---because, we know (either by Weierstrass, or just by inspection as you have done) a solution to $(*)$ exists while (i) and (ii) has no solution in $\{ g \leq 0 \}$.
Slater's condition is also a kind of constraint qualification. It is not related to complementary slackness. You can convert the original problem $(*)$ to one with equality constraint and apply Theorem of Lagrange, instead of KKT, but then it does not make sense to speak of Slater's condition.
It is also not correct to say that the problem is ill-posed.
A maximization problem is ill-posed if it does not have a solution. In this case, $(*)$ is clearly well-posed. The issue is applicability of KKT theorems.
In summary, inspecting the precise statements of various KKT theorems tells you KKT results do not apply to the given (counter-)example.
