Comments to the question (v2):

1. One nice property of the 2-norm (as compared to other norms, such as, e.g. the $p$-norm) is that it gives rise to an inner product via a so-called polarization trick. E.g. in the real case the polarization formula has 2 terms: $$ \langle u, v \rangle ~:=~\frac{1}{4} || u+v ||^2 -\frac{1}{4} || u-v ||^2 .$$ There is a similar 4-term polarization formula in the complex case. See also this related Phys.SE post.

2. It seems appropriate to mention that there exist metric theories, which are not based on Riemannian manifolds with a metric tensor and its corresponding 2-norm. One class of such generalized metric theories is Finsler geometry, see e.g. arXiv.org.

Comments to the question (v2):

1. One nice property of the 2-norm (as compared to other norms, such as, e.g. the $p$-norm) is that it gives rise to an inner product via a so-called polarization trick. E.g. in the real case the polarization formula has 2 terms: $$ \langle u, v \rangle ~:=~\frac{1}{4} || u+v ||^2 -\frac{1}{4} || u-v ||^2 .$$ There is a similar 4-term polarization formula in the complex case. See also this related Phys.SE post.

2. It seems appropriate to mention that there exist metric theories, which are not based on Riemannian manifolds with a metric tensor and its corresponding 2-norm. One class of such generalized metric theories is Finsler geometry, see e.g. arXiv.org.

Comments to the question (v2):

1. One nice property of the 2-norm (as compared to other norms, such as, e.g. the $p$-norm) is that it gives rise to an inner product via a so-called polarization trick. E.g. in the real case the polarization formula has 2 terms: $$ \langle u, v \rangle ~:=~\frac{1}{4} || u+v ||^2 -\frac{1}{4} || u-v ||^2 .$$ There is a similar 4-term polarization formula in the complex case. See also this related Phys.SE post.

2. It seems appropriate to mention that there exist metric theories, which are not based on Riemannian manifolds with a metric tensor and its corresponding 2-norm. One class of such generalized metric theories are Finsler geometry.

Comments to the question (v2):

1. One nice property of the 2-norm (as compared to other norms, such as, e.g. the $p$-norm) is that it gives rise to an inner product via a so-called polarization trick. E.g. in the real case the polarization formula has 2 terms: $$ \langle u, v \rangle ~:=~\frac{1}{4} || u+v ||^2 -\frac{1}{4} || u-v ||^2 .$$ There is a similar 4-term polarization formula in the complex case. See also this related Phys.SE post.

2. It seems appropriate to mention that there exist metric theories, which are not based on Riemannian manifolds with a metric tensor and its corresponding 2-norm. One class of such generalized metric theories is Finsler geometry, see e.g. arXiv.org.