Unless I am very badly mistaken it should hold that any solution $f:[0,\infty)\to \mathbb{R}^n$ ($n \in \mathbb{N}$) of the equation
$$f'(t)=g(f(t)) \quad \quad (t\in [0, \infty))$$ with a given initial condition $$f(0)=x_0$$
is unique (globally on all of $[0,\infty)$) provided that $g$ is a globally Lipschitz continuous function. However, I don't have an access to a library at the moment and surprisingly I had trouble finding a reference on Google even though it is a fairly basic result. I guess you can get the global uniqueness from the Picard-Lindelöf Theorem which gives me an interval on which the solution is unique whose length should only depend on the Lipschitz constant so I could then glue the unique solutions together to show global uniqueness...But I'd rather use a simple reference to the result I cite above.
Thanks for links to any references (preferably those that can be accessed online but inaccessible books are also OK, I can access a physical library, just not every day).
I don't know about online, but here's a book reference at least:
In Differential Equations with Applications and Historical Notes by Simmons, this is Theorem B in Section 70, Chapter 13 (what you call $t$ is called $x$ here):
“Let $f(x,y)$ be a continuous function that satisfies a Lipschitz condition $$ |f(x,y_1)-f(x,y_2)| \le K|y_1-y_2| $$ on a strip defined by $a \le x \le b$ and $-\infty < y < \infty$. If $(x_0,y_0)$ is any point of the strip, then the initial value problem $$ y'=f(x,y) ,\qquad y(x_0)=y_0 $$ has one and only on solution on the interval $a \le x \le b$.”
If you have a globally Lipschitz function (i.e., globally also in $x$, not just in $y$), you can apply this with an arbitrary interval $x \in [a,b]$ to deduce your result.
(The book includes the proof of the theorem, done from scratch with Picard iteration, almost but not exactly like for the local theorem.)