For existence theorem to be used f(x,y) needs to be continuous in the interval
\begin{equation}
R=\left\{(x, y):\left|x-x_{0}\right| \leq a,\left|y-y_{0}\right| \leq b\right\}, \quad(a, b>0)
\end{equation}
To find the interval actually you should randomly pick the area by yourself because you are analysing the equation and you pick those values of a and b according to the function so, you should pick some interval and test it with the existence and uniqueness theorems but they are not fully telling you the interval of validity so you should find the interval first according to some techniques.
In this particular problem, if you didn't immediately kind of see what was going on to be able to take a shortcut, you would start with this computation. For $x \neq y$:
$$\frac{|\sqrt{|x|}-\sqrt{|y|}|}{|x-y|}=\frac{||x|-|y||}{|x-y|(\sqrt{|x|}+\sqrt{|y|})} \\
\leq \frac{1}{\sqrt{|x|}+\sqrt{|y|}}.$$
You recognize that this is exact when $x,y$ have the same sign or just one of them is zero (so you haven't done some kind of blunt estimate that completely changes the behavior).
Therefore, how can you make $\sqrt{|x|}+\sqrt{|y|}$ very small? Make $y=0$ and then consider $x \to 0$. You do that and you conclude that your $f(t,x)$ is not Lipschitz in $x$ near zero.
This means that Picard-Lindelof can't be applied if the trajectory would pass through zero at some point. This may or may not imply non-uniqueness.
In this case, it gives a hint that there might be non-uniqueness for solutions that hit zero. And indeed you can find two solutions with say $x(0)=0$ by hand. One of them is identically zero, another is $\operatorname{sign}(t) t^2/2$ which can be seen by separation of variables. More generally there are actually infinitely many solutions to this problem, determined by the interval on which you choose to have $x=0$.
But this phenomenon is not determined solely by the regularity, it is also determined by the detailed behavior of the dynamics. In particular, this behavior does not occur for $x'=\sqrt{|x|}+1$ (which has exactly the same situation with respect to the hypotheses of Picard-Lindelof). In this case separation of variables works without any division by zero anywhere. The problem is that this "singularity" in $x'=\sqrt{|x|}$ requires the dynamics to become very slow as $x$ approaches zero (but not so slow that zero cannot be reached in finite time at all). This doesn't happen with $x'=\sqrt{|x|}+1$.
Generally, there is not really a nice necessary-and-sufficient condition for existence/uniqueness in ODEs. The local version of Picard-Lindelof works in most situations we frequently encounter, but when it is not applicable, we tend to need to do things on an ad hoc basis.
Best Answer
The function $f(y):=y^\alpha$ is not locally Lipschitz in $y=0$ for $\alpha \in (0,1)$. If we choose $\alpha$ in this interval and $y_0\neq 0$ then we can apply the Picard-Lindelöf theorem. The same theorem applies when $\alpha \geq 1$ since $y^\alpha$ is locally Lipschitz for all $\alpha \geq 1$ in all $y\in \mathbb{R}$.
When we lose the Lipschitz continuity we can construct $2$ solutions to prove non-uniqueness. For example if $\alpha=1/2$ and $y(1)=0$ we have the solutions $y\equiv 0$ and $y=\frac{(x-1)^2}{4}$. We can easily generalize this example to all $\alpha \in (0,1)$ and $y(x_0)=0$.