As I understand, the continuity of a function in a differential equation implies its existence. That is, even before talking about uniqueness, the differential equation $\frac{dx}{dt}=f(x)$ does not even HAVE solutions if its discontinuous.
However, consider the following $f(x)$:
$$f(x)= 2 x+2, x \geq 1$$
$$f(x)=2 x+1, x<1$$
Now, although such a velocity function would be PHYSICALLY impossible (as the sudden change in velocity would imply infinite acceleration), it seems to me that this solution solves the differential equation:
$$x(t)=\frac{2}{3} e^{2 t}-1, \quad \frac{\ln (3)}{2} \leq t<\infty$$ $$x(t)=\frac{1}{2} e^{2 t}-\frac{1}{2}, \quad 0 \leq t<\frac{\ln (3)}{2}$$
Sure enough, if we differentiate the top expression with respect to time, we get $2x+2=\frac{4}{3}e^{2t}$, and if we differentiate the bottom one with respect to time, we get $2x+1=e^{2t}$
Am I misunderstanding the theorem?
The solution of an ordinary differential equation is a continuously differentiable function. It is for this case that the proof methods in the existence theorems of Peano and Picard-Lindelöf-Cauchy-Lipschitz work.
As you can already see from your example, the solution that you constructed is not differentiable everywhere. It is a solution in a generalized sense. For broader application, you would have to define what this generalized sense is and what type of generalized non-continuous differential equations it applies to.
This has already been done, with Filippov's theory for ODE with a distributional right side. As https://en.wikipedia.org/wiki/Differential_inclusion claims, this theory still demands that solutions be continuous, and that that can still be weakened (and the resulting solutions still make sense).
Less general special cases are like your example systems that are smooth on sufficiently smoothly bounded compartments of the state space. The most interesting phenomenon to be observed there is the sliding mode where the vector fields on both sides of a boundary point inside the boundary. In a physical sense this would be the unstoppable object meeting an unmovable wall, in some extension of the solution one can consider the projection of the "joint forces" in direction of the boundary to provide a path to extend the solution.
The simplest example to examine is $\dot x=-{\rm sign}(x)$, an extension of that with more striking phase portraits is $\ddot x+a\,{\rm sign}(\dot x)+x=0$.