Is there a difference between one or two lines depicting the norm?

It really just varies according to the author/instructor. The only universal rule is that we use single bars for absolute values of real (and complex) numbers (e.g.$|-5|$). Once we start defining norms for other objects, we can choose single bars, double bars, or some other notation (although bars are very standard). In some contexts, we use $N(\alpha)$ to indicate a norm of $\alpha$. Reasons for using double bars (or any other notation) might include the desire to differentiate a vector norm, or some other norm, from the absolute value of a scalar.

If you are defining some kind of norm in your own writing, it's a good idea to define your notations before you use them, so that readers can follow your argument, even if they come from a context of using different notations.

(Single bars for absolute value is nearly universal. In some computer systems, however, absolute value of a real number $x$ is denoted $\mathrm{abs}(x)$. There may be other notations floating around, too.)

If I posted this as a question it would be a duplicate, so I decided to post it as an answer in the hope that it either gets confirmed or rejected - if the former, it will stay as a valid answer; if the later, it may stay as a counter-example, or be deleted.

Even though it seems as harping on $\vert \cdot \vert$ versus $\Vert \cdot \Vert$ is as pedantic as useless, there is some sense in which it reinforces the concept, and that is in the difference in the space of continuous functions between the absolute value of the function $\vert \cdot \vert,$ as in, for example for $f(x)=\sin x:$

$f(x) = \vert \sin x \vert$

and different norms, e.g.

$$\Vert f(x) \Vert_1 = \int_0^{2\pi}\vert \sin x \vert dx =4$$

vs

$$\Vert f(x) \Vert_2 = \left( \int_0^{2\pi}\vert \sin x \vert^2 dx \right )^{1/2} =\sqrt \pi$$

or

$$\Vert f(x) \Vert_\infty = \underset{0\leq x \leq 2\pi}{\max}\left(\vert \sin x \vert \right ) =1$$