Can a differential equation have non unique solutions?

Consider for example the equation $x' = 2\sqrt{|x|}$. For every $a$, the function $$ x_a(t) = \begin{cases} 0 & t < a \\ (t-a)^2 & t \ge a \end{cases} $$ is a solution. Note that for $a \ge 0$ all $x_a$ have $x_a(0) = 0$, so they are all solutions to the IVP $x' = 2\sqrt{|x|}, x(0) = 0$ and you usually discuss uniqueness for initial value problems, as otherwise uniqueness will almost never hold ($x' = 0$ has all constants as solutions).

Let your ODE be $y'-x\sqrt{y}=0, \; y(0)=0$. It is not difficult finding its solution on $\mathbb R$. It has at least two solutions as $y=0$ and $y=\frac{x^4}{16}$ passing through the origin. Can you see why the ODE has no unique solution?

Since you didn't specify an IVP, it is trivial to find an ODE with non unique solution, such as $y'(x)=0 \implies y(x)=C$, where $C\in\mathbb R$.