Example 23 Given an ellipse with eccentricity $e=\frac{2}{5} \sqrt{5}$, passing through the point $(1,0)$ and tangent to the line $l: 2 x-y+3=0$ at point $P\left(-\frac{2}{3}\right.$, $\left.\frac{5}{3}\right)$, with the major axis parallel to the $y$-axis. Find the equation of this ellipse.
Solve: Consider the point $P\left(-\frac{2}{3}, \frac{5}{3}\right)$ as a special case of the ellipse system $\left(x+\frac{2}{3}\right)^{2}+\frac{1}{5}\left(y-\frac{3}{3}\right)^{2}=k$ (i.e., when $k=0$, it is a "point ellipse"), then the circle system tangent to the line $l: 2 x-y+3=0$ at this point, is the ellipse system equation passing through the common point of the line $l$ and the "point ellipse":
$$
\left(x+\frac{2}{3}\right)^{2}+\frac{1}{5}\left(y-\frac{5}{3}\right)^{2}+\lambda(2 x-y+3)=0,
$$

Substituting the point $(1,0)$ into the above equation, we get $\lambda=-\frac{2}{3}$.
Therefore, the required ellipse equation is $x^{2}+\frac{1}{5} y^{2}=1$.