If we assume implicitly, that the polarisation is constant, we are allowed to skip this detail in or description, because the result does not change. However, if we assume that different polarisation components are present, we have to address each component separately -- assuming that you consider two orthogonal components, which do not mix.
By the way, this is also true for the wavelength of the light -- there exists no light source which possesses only a single wavelength (i.e. frequency) component --, the concept of a perfectly plane wave, or the idea of a medium with a homogeneous index of refraction. These are conceptional descriptions.
Here an example:
Suppose we'd like to describe a plane wave with "frequency" $\omega$, wave vector $\vec k = \frac{2\pi}{\lambda} \, \vec e_z$, and polarisation $\vec p = \frac{1}{\sqrt{2}}(\vec e_x + \vec e_y)$. We could either choose the cartesian coordinate system and write
$$
\vec E(\vec r, t)
% = E_0 e^{-i(\omega t - \vec k\cdot \vec r)} \; \vec p
= \frac{E_0}{\sqrt{2}} e^{-i(\omega t - k z)} \;
\begin{pmatrix}
1\\
1
\end{pmatrix}
$$
or we could rotate the reference frame by 45° and use the vector basis
$\{\vec e_{p_{\parallel}}, \vec e_{p_{\perp}}\}$, which leads to
$$
\vec E(\vec r, t)
= E_0 e^{-i(\omega t - k z)} \;
\begin{pmatrix}
1\\
0
\end{pmatrix}
$$
Now, if we assume that all electric fields are either parallel or anti-parallel to $\vec e_{p_{\parallel}}$, we know that we only have to consider the first component of the vector. Hence, by dropping the vector notation and implicitly only considering the first component of the vector, we are allowed to write
$$
E(\vec r, t)
= E_0 e^{-i(\omega t - k z)}
$$
Hence, the polarisation is no longer explicitly considered, but only implicitly.
