the $L^2$-norm of a signal is also applied as its energy!
Yes, the square of the $L_2$ norm of a signal is also by definition its energy $\mathcal{E}_x$.
There's nothing surprising, unbelievable, or mysterious in that though?
The concept of signal energy :
$$ \mathcal{E}_x = \int_{-\infty}^{ \infty } x(t)^2 dt\tag{1} $$
is fundamentally based on the concept of energy (or work) in physics as the Kinetic Energy of a particle with mass $m$ and velocity $v$ given by
$$ K = \frac{1}{2} m v^2 \tag{2}$$
There is also the concept of power defined as the rate of work $W(t)$ (or energy).
$$ p(t) = \frac{dW(t)}{dt}  \tag{3} $$
The relation between instantaneous power $p(t)$ and the total energy is
$$ \mathcal{E} = \int_{-\infty}^{\infty} p(t) dt \tag{4} $$
Electrical engineers ignore the mechanical roots, and rely on an electrical analog of energy as heat loss in an Ohmic resistor defined to be:
$$ \mathcal{E} = \int_{-\infty}^{\infty} p(t) dt \tag{5} $$
Where $p(t)$ is the instantaneous electric power associated with a current $i(t)$ passing through a linear time-invariant resistor $R$ , and is given by :
$$ p(t) = R  \cdot i^2(t) \tag{6} $$
( $p(t) = v^2(t)/R $ is also an equivalent expression, based on Ohm's law $v(t) = R i(t)$)
Then the energy of the current signal passing though a linear time invariant system (denoted by simple Ohmic resistor $R$) is given by :
$$ \mathcal{E} = \int_{-\infty}^{\infty} R \cdot i^2(t) dt \tag{7}$$
Now at this point, just ignoring the resistor $R$ (or setting it to $R=1$), and replacing the current variable with a general unitless $x(t)$, we arrive at the mathematical definition of signal energy of as:
$$ \mathcal{E} = \int_{-\infty}^{\infty} x^2(t) dt \tag{8}$$
That being clear, in a parallel course, the study of normed linear Hilbert spaces also consider mathematical p-th Euclidean norm of a complex valued vector as :
$$ L_p = \left( \int_{-\infty}^{\infty} |x(t)|^p dt \right)^{1/p} \tag{9}$$
And you can see that the square of the case $p=2$ corresponds to the signal energy as defined in Eq.(8).
All of these can also be transferred to discrete-time domain.