Christopher Brierley Jones describes the process on his website how an ellipse can be fit perfectly into any convex quadrilateral. The process is very similar to what DMGregory describes in his answer to extract the ellipse parameters from the matrix representation.
The key insight is that instead of projecting the ellipse directly, you can project the bounding rectangle. That will always result in an convex quadrilateral. Then you can use Jones process to fit an ellipse into the resulting quadrilateral.
You already mentioned that the axes are no longer aligned after the projection. Note that not even the center of the resulting ellipse aligns with the projected center of the circle. Jones describes the relation between the original circle and the projected ellipse as "inscrutable".
I implemented function for mapping a convex quad to an {cx,cy,ra,rb,angle} ellipse in JavaScript.
// W,X,Y,Z are {x:number,y:number} vertices of the convex qudriliteral.
function quadToEllipse(W,X,Y,Z) {
// Reconstruct matrix that transforms the unit square ((-1,-1), (1,1)) into quad (W,X,Y,Z)
const m00 = X.x * Y.x * Z.y - W.x * Y.x * Z.y - X.x * Y.y * Z.x + W.x * Y.y * Z.x -
W.x * X.y * Z.x + W.y * X.x * Z.x + W.x * X.y * Y.x - W.y * X.x * Y.x;
const m01 = W.x * Y.x * Z.y - W.x * X.x * Z.y - X.x * Y.y * Z.x + X.y * Y.x * Z.x -
W.y * Y.x * Z.x + W.y * X.x * Z.x + W.x * X.x * Y.y - W.x * X.y * Y.x;
const m02 = X.x * Y.x * Z.y - W.x * X.x * Z.y - W.x * Y.y * Z.x - X.y * Y.x * Z.x +
W.y * Y.x * Z.x + W.x * X.y * Z.x + W.x * X.x * Y.y - W.y * X.x * Y.x;
const m10 = X.y * Y.x * Z.y - W.y * Y.x * Z.y - W.x * X.y * Z.y + W.y * X.x * Z.y -
X.y * Y.y * Z.x + W.y * Y.y * Z.x + W.x * X.y * Y.y - W.y * X.x * Y.y;
const m11 = -X.x * Y.y * Z.y + W.x * Y.y * Z.y + X.y * Y.x * Z.y - W.x * X.y * Z.y -
W.y * Y.y * Z.x + W.y * X.y * Z.x + W.y * X.x * Y.y - W.y * X.y * Y.x;
const m12 = X.x * Y.y * Z.y - W.x * Y.y * Z.y + W.y * Y.x * Z.y - W.y * X.x * Z.y -
X.y * Y.y * Z.x + W.y * X.y * Z.x + W.x * X.y * Y.y - W.y * X.y * Y.x;
const m20 = X.x * Z.y - W.x * Z.y - X.y * Z.x + W.y * Z.x - X.x * Y.y + W.x * Y.y + X.y * Y.x - W.y * Y.x;
const m21 = Y.x * Z.y - X.x * Z.y - Y.y * Z.x + X.y * Z.x + W.x * Y.y - W.y * Y.x - W.x * X.y + W.y * X.x;
const m22 = Y.x * Z.y - W.x * Z.y - Y.y * Z.x + W.y * Z.x + X.x * Y.y - X.y * Y.x + W.x * X.y - W.y * X.x;
// invert matrix
const determinant = +m00*(m11*m22-m21*m12) -m01*(m10*m22-m12*m20) +m02*(m10*m21-m11*m20);
if(determinant == 0) return null;
const invdet = 1/determinant;
const J = (m11*m22-m21*m12)*invdet;
const K = -(m01*m22-m02*m21)*invdet;
const L = (m01*m12-m02*m11)*invdet;
const M = -(m10*m22-m12*m20)*invdet;
const N = (m00*m22-m02*m20)*invdet;
const O = -(m00*m12-m10*m02)*invdet;
const P = (m10*m21-m20*m11)*invdet;
const Q = -(m00*m21-m20*m01)*invdet;
const R = (m00*m11-m10*m01)*invdet;
// extract ellipse coefficients from matrix
const a = J*J + M*M - P*P;
const b = J*K + M*N - P*Q;
const c = K*K + N*N - Q*Q;
const d = J*L + M*O - P*R;
const f = K*L + N*O - Q*R;
const g = L*L + O*O - R*R;
// deduce ellipse center from coefficients
const centerX = (c*d - b*f) / (b*b - a*c);
const centerY = (a*f - b*d) / (b*b - a*c);
// deduce ellipse radius from coefficients
const radiusA = Math.sqrt(2*(a*f*f + c*d*d + g*b*b - 2*b*d*f - a*c*g)/((b*b - a*c) * (Math.sqrt((a-c)*(a-c) + 4*b*b) - (a+c))));
const radiusB = Math.sqrt(2*(a*f*f + c*d*d + g*b*b - 2*b*d*f - a*c*g)/((b*b - a*c) * (-Math.sqrt((a-c)*(a-c) + 4*b*b) - (a+c))));
// deduce ellipse rotation from coefficients
let angle = 0;
if(b==0 && a <= c) {
angle = 0;
} else if(b == 0 && a >= c) {
angle = Math.PI / 2;
} else if(b != 0 && a > c) {
angle = Math.PI / 2 + 0.5 * (Math.PI / 2 - Math.atan2((a-c), (2*b)));
} else if(b != 0 && a <= c) {
angle = Math.PI / 2 + 0.5 * (Math.PI / 2 - Math.atan2((a-c), (2*b)));
}
return {
centerX, centerY, radiusA, radiusB, angle
}