There is no reason to expect a unique solution to a system of simultaneous non-linear equations. Let's take a very simple example.

$y = ax^2 + c,\ where\ a \ne 0\ and\ a,\ c \in \mathbb R.$

$y = dx^2 + f,\ where\ d \ne 0,\ d \ne a,\ f \ne c,\ and\ \ d,\ f \in \mathbb R.$

Find the conditions for real solutions to the system above.

$0 = y - y = ax^2 + c - (dx^2 + f) = (a - d)x^2 + (c - f) \implies$

$x = \pm\ \sqrt{\dfrac{f - c}{a - d}}.$

$No\ real\ solutions\ if\ \dfrac{f - c}{a - d} < 0.$

$Two\ real\ solutions\ if\ \dfrac{f - c}{a - d} > 0.$

This system will have either two real solutions or no real solution and will never have a unique real solution.