If $x^2 + y^2 = 4$ represents the section of a cone by a plane horizontal to the cone, a circle, and $y^2 - x^4 = 4$ represents the section of a cone by a plane vertical to the cone, a hyperbola, what do $x^3 + y^3 = 8$ and $y^3 - x^3 = 8$ represent? sections of a 4-dimensional version of a cone by a 3-dimensional object? What about equations like $x^3 + y^2 = 8$? What does x + y = 2 represent (my first thought was the section of a triangle by a line, but that doesn't seem to make much sense)? Do these (and by extension all equations) represent a section of one object of n dimension by another object of n-1 dimension?
Hopefully you can answer all of these questions, I put the title as being Do all equations represent a section of an n dimensional object by another object of n-1 dimension? because I think it covers all of my questions in a broader way.
In order to get started, let's look at your first example. If you're working in a two-dimensional Cartesian space, the equation $x^2 + y^2 = 4$ is a circle. Of course in two-dimensional Cartesian space you cannot make a three-dimensional figure at all, since there's no third dimension to extend into.
In three-dimensional Cartesian coordinates the equation $x^2 + y^2 = 4$ describes an infinite cylinder. The $z$ coordinate can be anything (since it was not mentioned in the equation) but the equation restricts $x$ and $y$ so that the point $(x,y,z)$ is exactly $2$ units away from the $z$-axis.
In three dimensions, if you want to describe a circle, you need two equations. One equation can just select a $z$ coordinate. The simultaneous equations $x^2 + y^2 = 4$ and $z = 2$ select a circle centered on the $z$-axis in a plane parallel to the $x,y$ plane and at a distance $2$ from the $x,y$ plane. (You can also get a circle without setting $z$ to a constant; this is merely one way to get a circle.)
The circle given by $x^2 + y^2 = 4$ and $z = 2$ is a section of the cone $x^2 + y^2 - z^2 = 0,$ consistent with your question. But it's also (trivially) a section of the cylinder $x^2 + y^2 = 4,$ a section of the sphere $x^2 + y^2 + (z - 2)^2 = 4,$ a section of the sphere $x^2 + y^2 + z^2 = 8,$ and a section of the hyperboloid $x^2 + y^2 - \frac12 z^2 = 2.$
Somewhat more generally, you can take any plane shape described by two simultaneous equations, a polynomial equation in $x$ and $y$ and an equation setting $z$ to a constant, delete the "$z=\text{constant}$" equation, and you have the equation of a kind of cylinder--not generally a circular cylinder, but these other shapes are generically called cylinders as well. If the original equations included $z=0,$ you could get a more interesting shape by deleting $z=0$ and inserting some $z$ terms (any positive power of $z$ multiplied by any constant, any non-negative power of $x,$ and any non-negative power of $y$) into the polynomial equation so it is a polynomial equation of $x,$ $y,$ and $z$ that reduces to the original polynomial equation when $z=0.$ If the original equations included $z=c,$ instead of inserting $z$ terms in the equation, insert terms using positive powers of $z - c$ multiplied by any constant, any non-negative power of $x,$ and any non-negative power of $y$. Then your original plane figure will be a section of the three-dimensional figure described by your new equation.
What will the three-dimensional figure be? It could be any number of shapes. You can easily throw enough polynomial terms into the equation to produce a shape that has never been given a name. And if you already had an unnamed shape in two dimensions then it almost sure has no name in three.
The same principles apply to $n$ dimensions. Set the last dimension constant and a polynomial equation over the other $n-1$ coordinates describes a figure in $n-1$ dimensions. This could be a section of any one of an infinite (mostly unnamed) $n$-dimensional figures.
If the equations can be general equations over the coordinates then the possibilities open up even more. But that only makes it even harder to categorize the results.