The Euclidean norm is often used based on the assumption that the Euclidean distance of two points is a reasonable measure of distance. But unless this is the case, this choice is not preferable to a problem-adapted choice. For example if the typical size of the components of a vector are very different (since they mean very different things), the Euclidean norm is very poor as it hardly takes into account the effects of changes in the small-size components.
In such a case, one either needs to first scale the vectors to have similar sized components before applying norms, or one must use a norm that scales different components differently.
The norm $\|x\|$ of a vector $x$ (and similarly for matrices and functions) is a measure of its size; this measure must be adapted to the meaning of the problem you are solving. In finite dimensions, all norms are equivalent, in the sense that they describe the same topology; but the numerical values may depend quite a lot on the particular norm. (For the topology, the only thing of interest is the limit. In finite D this is independentent of the norm, i.e., $\|x_k−x\|\to 0$ in any norm if $\lim x_k=x$. But how close one is to the limit depends a lot on in which norm you measure it.)
Therefore one must choose a meaningful norm to get meaningful results.
In infinite-dimensional spaces (which in particular includes the common function spaces), norms are no longer equivalent, and different norms may lead to different topologies. Now one must choose a suitable norm even to get finite results, and bounding terms may be impossible wwithout a good choice of the norm.
As an exercise, I'd like to suggest that you compare the values of the $p$-norm for $p=1,2,\infty$ for a variety of vectors in $R^n$ parameterized by $n$, and do the same in various spaces of sequences $x=(x_1,x_2,\dots)$. You'll then appreciate the differences. A good example is the vector with $i$the entry $x_i=\epsilon/i^s$, where $s>0$. Here for tiny $\epsilon$ and large $n$ (approximate the sum by an integral)
$\|x\|_p\approx \epsilon\frac{1-1/n^{ps-1}}{ps-1}$, which becomes infinitely large as $n\to\infty$ when $p\le 1/s$ but remains tiny when $p>1/s$.