I know how to use the Cayley-Klein metric for lines in a hyperbolic plane, more specifically in the Beltrami-Klein model. But how is the Cayley-Klein metric used in a Euclidean plane as Wikipedia suggests that it could?
"The Cayley–Klein metrics are a unifying idea in geometry since the method is used to provide metrics in hyperbolic geometry, elliptic geometry, and Euclidean geometry."
Is it through the cross-ratio when there is no projective transformation at all?
You can use the Cayley-Klein metric for angles in hyperbolic geometry. The fundamental quadric there would be chosen in such a way that the dual view of it consists of the two ideal circle points $[1:\pm i:0]$. So the operation of constructing the tangents in a given point reduces to joining that point to these two circle points, resulting in a pair of complex conjugate lines. Together with the original lines this gives you a cross ratio of four lines which has absolute value one, so that it's logarithm is purely imaginary and can be scaled to a regular angle.
If you try to measure distances, not angles, then you need the primal view of the fundamental quadric, which is the line at infinity with multiplicity two. So the two intersections with the conic would always be the same. Using the notation from the Wikipedia article, $p=q$. That makes numerator equal denominator so the cross ratio becomes one, its logarithm zero and all distances between finite points would evaluate as zero. You don't have an absolute distance metric in Euclidean geometry.
There are variations of the formulas which can be used for computing the ratio between distances. Some of them (e.g. derived from limit considerations) avoid the zeros. These formulas allow for a relative metric, where you can say that one distance is twice as much as another distance even though both of them would be zero as absolute measurements. By defining one distance to be the unit of length, you can establish length measurements there, but everything will then depend on the distance you're using as reference.
I would recommend chapters 20 and 21 of Jürgen Richter-Gebert's Perspectives on Projective Geometry if you want to read more on that.