If $f(x, y, z)$ is a differentiable function (with the derivative satisfying some special conditions*) of 3 variables x, y, z (it can end up not depending on some of those variables), then the equation $f(x, y, z) = c$ defines a 2-dimensional "manifold" (think of it like a curvy sheet of paper).
In general, if f is a function of n variables, then f(those n variables) = c defines an n-1 dimensional manifold.
A plane is a 2 dimensional manifold, which is why you can express it the way you did. A line is a one dimensional manifold, which is why we can't just express it with something like $ax+by+cz=d$—we need more equations to reduce the dimension of the manifold it defines. (hence the vector formulation!)
A good way to think about manifolds is that the number of variables are the degrees of freedom, and the number of equations are the constraints, and the dimension of the shape you're producing is degrees of freedom minus constraints.
For a plane in 3-dimensional space, you have 3 variables and one equation. 3-1=2, which is the dimension of a plane. For a line in 3-space, we have 3 variables, but a line is one-dimensional, so one equation does not suffice to describe it.
*If you're interested, this "special condition" is surjectivity. If we have this property at a point, then the point set is locally a manifold.
**If you're really interested, look into how one can use something called the implicit function theorem (which is, among other things, a way of stating the relationship between degrees of freedom and constraints) to define manifolds. Vector Calculus, Linear Algebra, and Differential Forms by Hubbard & Hubbard gives an excellent and accessible treatment of the subject.