This is subtle. It is not at all clear a priori what the correct generalization of "linear independence" is. This is because polynomials can nontrivially divide each other; for example, the system in two variables
$$x - y = 0$$
$$x^2 - y^2 = 0$$
has infinitely many solutions $\{ (x, y) : x = y \}$, and the first equation divides the second but they are linearly independent. As a more complicated example, consider the system in three variables
$$x = 0$$
$$y = 0$$
$$xz - yz = 0$$
which still has infinitely many solutions $\{ (x, y, z) : x = y = 0 \}$ ($z$ is arbitrary). This time none of the equations divide each other but the third one is a linear combination of the first two with polynomial coefficients.
From the modern perspective, coming from commutative algebra and/or algebraic geometry, we can understand a system of polynomial equations by understanding the ideal they generate in the ring of polynomials, and one plausible guess as to the appropriate generalization of "linear independence" is most clearly stated in this language: it's that if we have $n$ polynomials $f_1, \dots f_n$ then the ideal $I = (f_1, \dots f_n)$ they generate cannot be generated by fewer than $n$ polynomials. This is related to the notion of a complete intersection except I have only seen complete intersections discussed in projective space. There is also the related notion of a regular sequence which I unfortunately know very little about.
This is still not enough to guarantee a unique solution. Generically one now expects $(\deg f_1) (\deg f_2) \dots (\deg f_n)$ solutions, at least over $\mathbb{C}$. A very simple example is
$$x - y = 0$$
$$x - y^2 = 0$$
which is equivalent to $y = y^2$ and hence which has two solutions $(1, 1)$ and $(0, 0)$. For more complicated examples see the Wikipedia article on Bezout's theorem: there are nice geometric examples coming from intersections of conic sections.
I don't know off the top of my head if "the ideal is not generated by fewer than $n$ polynomials" is enough to guarantee finitely many solutions. This is probably extremely classical and I hope someone else comes along who does know.
Another plausible generalization of "linearly independent" is that the Jacobian of $f_1, \dots f_n$ is generically invertible; equivalently, the differentials $df_1, \dots df_n$ are generically linearly independent. ("Invertible everywhere" is a quite strong condition.) We should be able to conclude something using the inverse function theorem from this but unfortunately my command of the subject is not quite strong enough to tell immediately what.
