From ellipse equation to circular cone axis

From a standard ellipse equation in 2D $ax^2 + bxy + cy^2 + dx + ey + f = 0$, is it possible to retrieve the axis of the corresponding 3D circular cones? I sense there an infinity of possible circular cones, but only two possible axis. However I can't find any information on how to find these axis. I looked into Dandelin spheres but couldn't find any methods on how to construct them from an ellipse.

One can use the results found for this similar question to obtain what you want in the particular case of an ellipse with equation $$ {x^2\over a^2}+{y^2\over b^2}=1, $$ with $a\ge b$. In this case you can take as vertex of the cone any point on the hyperbola in the $xz$-plane with equation $$ {x^2\over a^2-b^2}-{z^2\over b^2}=1. $$ In the diagram below I'm showing an example with $a=3$ and $b=2$. The axis of the cone is the bisector of $\angle BAC$, where $A$ is the cone vertex while $BC$ is the major axis of the ellipse.

The construction using Dandelin spheres is not very complicated.

First you place a sphere $S$ so that it is tangent to the plane of the ellipse at one focus of the ellipse. The radius of the sphere $S$ must not equal the semiminor axis of the ellipse; any other radius is OK.

Now consider the plane $\pi_1$ passing through the two foci of the ellipse and the center of the sphere $S.$ The plane $\pi_1$ is perpendicular to the plane of the ellipse and contains the major axis of the ellipse. The intersection of the sphere $S$ with the plane $\pi_1$ is a circle, $C.$

There are threee lines in the plane $\pi_1$ that pass through at least one end of the major axis of the ellipse and are tangent to the circle $C.$ One of those lines is the line on which the major axis lies. The other two lines intersect at a point $P_0.$

The point $P_0$ is the apex of a right circular cone such that the ellipse is the intersection of that cone with the plane containing the ellipse, and the sphere $S$ is one of the Dandelin spheres of that cone and the ellipse.

Note that if the radius of the sphere were equal to the semiminor axis, the two lines would be parallel. Instead of giving you the apex of a cone, they would lie on a right circular cylinder such that the intersection of that cylinder with the plane containing the ellipse.

As a consequence of this construction, you can prove that the difference in distances from $P_0$ to the two ends of the ellipse's major axis is equal to the distance between the foci of the ellipse. Therefore $P_0$ lies on a hyperbola whose foci are the vertices of the ellipse (the ends of the major axis of the ellipse) and that passes through the foci of the ellipse, as shown in Aretino's answer.