I have seen both $\left|\vec{u}\right|$ and $\left\|\vec{u}\right\|$ when referring to the Euclidean length of a geometric vector $\vec{u}$. Which notation is preferred. Is it true that the latter should be reserved for function norms like
$$ \left\|\vec{u}\right\| =\int_{\mathbb{R}^3}\left|\vec{u}\left(\vec{r}\right)\right|^2\,\mathrm{d}V $$
which is the $L^2$ norm of the vector field $\vec{u}$ on $\mathbb{R}^3$
Yes, both are preferred. There are various considerations. For example, some people use $\vert$ for set notation like $\{x\in\mathbb{R}\,\vert\,x<0\}$. Then $\Vert$ helps to avoid ambiguity. Sometimes $\vert$ is for an absolute value of an element of a simple algebraic structure such as a field, whereas $\Vert$ is then used for higher structures, like matrices, functions, operators etc. That's how you have used it. But a single symbol is generally unambiguous because the meaning is argument-class-dependent. You just need to look at the argument to see if it is a real number, complex number, function, matrix etc., and then you know how to interpret $\vert$ or $\Vert$. Thus the distinction between $\vert$ and $\Vert$ is largely a hint to help the reader see the meaning more easily.