What are the initial conditions for this coupled advection/transport system?

Consider the coupled transport (i.e. advection) system

\begin{align} \partial_t u + b\partial_x \phi &= 0,\\ \partial_t \phi + b\partial_x u &= 0, \end{align}

where $$u(x,t),\phi(x,t) \in C^2\left([0,1]\times \mathbb{R}^+\right)$$ are unknown, and $$b\in \mathbb{R}$$ is given and constant.

By combining the equations, one can recover two classical, decoupled, wave equations:

$$\partial_{tt} u - b^2 \partial_{xx} u = 0$$

and similar for $$\phi$$. The latter accepts one initial condition for the unknown $$u/\phi$$, and another one for its time derivative $$\partial_t u / \partial_t \phi$$. The reason for this property - to my understanding - is the structure of d'Alembert's solution, see e.g. https://mathworld.wolfram.com/dAlembertsSolution.html for a derivation.

How can I use this knowledge to find the "correct" initial conditions for the coupled transport system? How many conditions are required and which?

Intuitively, I would think that I need one initial condition for $$u$$ and one for $$\phi$$, as the time derivatives are first order. Is this correct?

The coupled transport system and the two wave equations are analytically equivalent, as far as I understand. Can I recover the very initial conditions I have to choose for the coupled transport system, such that the problems are equivalent (based on the chosen initial conditions for the wave equation)?

You are correct: for the system of first order PDEs \begin{align} \partial_t u + b\partial_x \phi &= 0, \tag{1a} \\ \partial_t \phi + b\partial_x u &= 0, \tag{1b} \end{align} you need one initial condition for $$u$$ and one for $$\phi$$, specifically \begin{align} u(x,0) &= u_0(x), \tag{2a} \\ \phi(x,0) &= \phi_0(x). \tag{2b} \end{align} On the other hand, for the second order PDE $$\partial_{tt} u - b^2 \partial_{xx} u = 0, \tag{3}$$ you need two initial conditions, $$u(x,0)$$ and $$\partial_t u(x,0)$$. The first is provided by $$(2\text{a})$$, and the second can be derived from $$(1\text{a})$$ and $$(2\text{b})$$: $$\partial_t u(x,0)=-b\partial_x\phi(x,0)=-b\phi_0'(x). \tag{4}$$