[Christopher Brierley Jones describes the process on his website][1] 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".

[![enter image description here][2]][2]

I implemented function for mapping a convex quad to an `{cx,cy,ra,rb,angle}` ellipse in JavaScript.

[You can see a live demo here][3]

```
// 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
	}
```

*Update*: [See this link][5] for an example where this technique is used for drawing ellipses onto the faces of a rotating 3d cube as shown below:

[![ellipses on the sides of a 3d cube][4]][4]


  [1]: http://chrisjones.id.au/Ellipses/ellipse.html
  [2]: https://i.sstatic.net/xTTVQ.png
  [3]: https://static.laszlokorte.de/quad/
  [4]: https://i.sstatic.net/FkOb9.png
  [5]: https://bafybeieblkhw6vmpeonreifsg2iaoj7l777mcdqedul5xodlthkpufsmky.ipfs.dweb.link/