To solve this problem, we examine the vector $\overrightarrow{AB}$ and its relationship with the coordinate planes.

Given points $A(1,3,0)$ and $B(0,3,-1)$, we can calculate the vector $\overrightarrow{AB}$ as follows:
$$\overrightarrow{AB} = B - A = (0-1, 3-3, -1-0) = (-1,0,-1).$$

The next step is to consider the orientation of $\overrightarrow{AB}$ relative to the various coordinate planes.

A normal vector to the plane $xOz$ is $\overrightarrow{n}=(0,1,0)$. To determine if $\overrightarrow{AB}$ is parallel or perpendicular to the $xOz$ plane, we compute the dot product of $\overrightarrow{AB}$ and $\overrightarrow{n}$:
$$\overrightarrow{AB}\cdot \overrightarrow{n} = (-1,0,-1) \cdot (0,1,0) = (-1)\cdot0 + 0\cdot1 + (-1)\cdot0 = 0.$$

Since the dot product is $0$, it means $\overrightarrow{AB}$ is perpendicular to the normal vector $\overrightarrow{n}$ of the plane $xOz$. However, being perpendicular to the normal vector implies that $\overrightarrow{AB}$ is not perpendicular to the plane itself but rather parallel to it, as it does not lie within the plane $xOz$ and does not intersect it in a manner that would be considered perpendicular.

Therefore, the correct conclusion is that line $AB$ is parallel to the coordinate plane $xOz$.

Hence, the answer is: $\boxed{C}$.