In all the references I checked the standard initial value problem for an ODE is stated as: \begin{equation} \begin{cases} y'=F(y,t)\\ y(t_0)=y_0 \end{cases} \end{equation} for some $F:\mathbb{R}^{n+1}\to\mathbb{R}^n$, and under hypotheses on $F$ one can conclude existence, uniqueness, etc.
My question is: is there a standard way to address the problem: \begin{equation} \begin{cases} y'=F(y,t)\\ y_i(t_i)=y_{0,i}, \quad i=1,\ldots, n \end{cases} \end{equation} where the "initial condition" is not given for the same $t_0$ for all the components of $y$, but at different $t_is$? where by address I mean, to begin with, check existence and uniqueness conditions.
The equation $y^{\prime\prime}=-y$ has general solutions $y=C\sin(t)+D\cos(t)$. Even though your question is about first order ODE, since it allow $y$ to be vector we can convert this problem into a first order vector ODE where the components are $y$ and $\dot{y}$
So the initial value problem $y(0)=0$ has general solutions $y=C\sin(t)$. In particular, for all such solutions, $y^{\prime}(\frac{\pi}{2})=0$. When converted to the first order vector ODE, this mean $\dot{y}(\frac{\pi}{2})=0$. So the initial value problem with different starting points $y(0)=0,\dot{y}(\frac{\pi}{2})=1$, in particular, is unsolvable.
Hopefully this explains why existence of solution is not a simple matter when you have different starting points for different coordinates.