I am totally confused, take the ODE $$u-tu'-(u')^2=0$$ Clearly $-t^2/4$ and $ct+c^2$ are both solutions, and if we take $c = 0$, then they both satisfy the initial condition $u(0) = 0$.
Why does this not contradict uniqueness?
Addendum: The Existence and Uniqueness Theorem.
Consider the initial value problem $X' = F(X)$, $X(0) = X_0$ where $X_0 ∈ \Bbb R^n$. Suppose that $F : \Bbb R^n → \Bbb R^n$ is $C^1$. Then there exists a unique solution of this initial value problem. More precisely, there exists $a > 0$ and a unique solution $X : (−a, a) → \Bbb R^n$ of this differential equation satisfying the initial condition $X(0) = X_0$.