$\newcommand{\Reals}{\mathbf{R}}$Euclidean plane geometry can be constructed analytically by starting with the Cartesian plane $(\Reals^{2}, +, \cdot)$ (the set of ordered pairs of real numbers with the usual operations of vector addition and scalar multiplication) and equipping it with the distance function coming from the Pythagorean theorem,
$$
d\bigl((x_{1}, x_{2}), (y_{1}, y_{2})\bigr)
= \sqrt{(y_{1} - x_{1})^{2} + (y_{2} - x_{2})^{2}},
\tag{1}
$$
or $(\Delta s)^{2} = (\Delta x)^{2} + (\Delta y)^{2}$.
A point is an element of $\Reals^{2}$.
A line is the zero locus of a first-degree polynomial, $ax + by = c$ with $(a, b) \neq (0, 0)$.
Two lines $a_{1}x + b_{1}y = c_{1}$ and $a_{2}x + b_{2}y = c_{2}$ are parallel if $a_{1}b_{2} - a_{2}b_{1} = 0$.
Two lines $a_{1}x + b_{1}y = c_{1}$ and $a_{2}x + b_{2}y = c_{2}$ are perpendicular if $a_{1}b_{1} + a_{2}b_{2} = 0$.
A circle is the locus of $(x - x_{0})^{2} + (y - y_{0})^{2} = r^{2}$.
A rigid motion is a transformation of the form
$$
\left[\begin{array}{@{}c@{}}
x \\
y \\
\end{array}\right] \mapsto
\left[\begin{array}{@{}rr@{}}
C & -S \\
S & C \\
\end{array}\right]\left[\begin{array}{@{}c@{}}
x \\
y \\
\end{array}\right] + \left[\begin{array}{@{}c@{}}
x_{0} \\
y_{0} \\
\end{array}\right],\quad
C^{2} + S^{2} = 1.
$$
Two angles are congruent if there exists a rigid motion taking one to the other.
And so forth. (These claims are doubtless familiar, but it requires rather extensive work to establish carefully that they model the Euclidean postulates. Patrick Ryan's Euclidean and Non-Euclidean Geometry, for example, is a nice reference.)
If you'll forgive my putting words in the mouths of great mathematicians:
To Riemann, the Euclidian plane is not merely a rigid, algebraic structure. Instead, we may think of the Euclidean plane as having a tangent space at each point.
Thanks to parallelism (the existence of a "global compass", the orthonormal frame whose value at each point is the standard basis of $\Reals^{2}$), any two tangent spaces to the Euclidean plane may be canonically identified. The tangent bundle therefore trivializes canonically as soon as we pick a distinguished orthonormal basis at one point. (This is tantamount to introducing Cartesian coordinates in the Euclidean plane, i.e., to fixing an origin, a distinguished pair of oriented lines, and the metric (1) with respect to the resulting coordinate system.)
The Pythagorean theorem now acquires an infinitesimal character:
$$
ds^{2} = dx^{2} + dy^{2},
$$
which is precisely Lanczos's equation (15.7) with $g_{ik} = \delta_{ik}$, the Kronecker delta.
In Riemann's view, for every ordered triple $(E, F, G)$ of (suitably smooth) functions with $EG - F^{2} > 0$ everywhere, we may consider the "variable" inner products
$$
g = E\, dx^{2} + F\, (dx\, dy + dy\, dx) + G\, dy^{2},
$$
and use these to do "infinitesimally Euclidean geometry", in the sense that each tangent space acquires the structure of the Euclidean plane, but these structures vary (smoothly) from point to point.
Riemann therefore sees Euclidean plane geometry as something like a two-dimensional continuum of Euclidean geometries that happen to be globally mutually-consistent (from the perspective of internal measurements).
The "certain class of functions" Lanczos mentions presumably refers to ordered triples $(E, F, G)$ for which:
The Gaussian curvature vanishes (a second-order fully-non-linear PDE, see the Brioschi formula).
The metric is complete (geodesics are defined for all time) and the universe simply-connected (to avoid topological issues, such as cylinders and flat tori, in which geodesics extend forever, but not the way they "should" for Euclidean geometry).
The functions $E = G = 1$, $F = 0$ are the simplest example. The most general example is to let $\Phi:\Reals^{2} \to \Reals^{2}$ be a diffeomorphism. The functions
$$
E = \Phi_{x} \cdot \Phi_{x},\qquad
F = \Phi_{x} \cdot \Phi_{y},\qquad
G = \Phi_{y} \cdot \Phi_{y}
$$
also define Euclidean geometry, via the global change of coordinates $\Phi$.
