A single equation in the three variables $x, y$, and $z$ constitutes a single constraint on the freedom of the point $P = (x, y, z)$ to lie anywhere in $\mathbb{R}^3$. Does such a constraint guarantee that $P$ lies on a two-dimensional surface?
Edit
Clearly, it is possible for an equation in three variables to not represent a surface in $\mathbb{R}^3$. For example, the equation $x^2 + y^2 + z^2 = 0$ represents the origin, and $x^2 + y^2 + z^2 = -1$ represents the empty set.
Is it possible for a single equation in three variables to impose no constraints on the freedom of a point?
If you restrict what types of equations you are looking at, the answer is yes. If you don't include such a restriction, then the answer is no.
The subspace of $\Bbb R^3$ given by $\{(x,y,z)\in\Bbb R^3~:~ax + by + cz = 0\}$ where $a,b,c$ are real numbers with at least one of them non-zero will indeed always describe a two-dimensional subspace of $\Bbb R^3$ and describes a plane passing through the origin.
The subset of $\Bbb R^3$ given by $\{(x,y,z)\in\Bbb R^3~:~ax+by+cz=d\}$ where at least one of $a,b,c$ is nonzero and $d$ is nonzero is a two-dimensional affine space and describes a plane not passing through the origin.
If you were to consider other possible equations in forms different than $ax+by+cz=d$ then it will depend on what the equation is. $x^2+y^2+z^2=0$ describes the origin only. $x^2+y^2+z^2=1$ describes the unit sphere, $0=0$ describes the entirety of $\Bbb R^3$, etc...