How to make a shape layer responsively adapt to the length of two different text layers?

You need to use sourceRectAtTime height and width for both text layers and add them together, plus the padding between the lines for the height, and use the longest of the two layers for width plus padding. This will give you the size of the shape layer background.

Then you need to tie the position of the second text layer, your subtitle, by adding the padding and the height + the top for vertical position and then use width + left + padding to position the shape layer horizontally. 

If you use index - 2 to define the first text layer and index - 1 to define the second text layer and put the shape layer below both text layers, these expressions will do the trick:

//Shape Layer/Contents/Rectangle 1/Size
//source layers 
src1 = thisComp.layer(index - 2);
src2 = thisComp.layer(index - 1);
// text size
tBox1 = src1.sourceRectAtTime();
tBox2 = src2.sourceRectAtTime();
// max width
tWidth = Math.max(tBox1.width, tBox2.width).toFixed(2);
//  layer spacing 
lyrSpace = effect("Leading Middle")("Slider");
// total height
srcHeight = Math.ceil(tBox1.height + tBox2.height);
tHeight = srcHeight + lyrSpace;
// box size
txtBase = [tWidth, tHeight];
// padding control
cornerPad = content("Rectangle 1").content("Rectangle Path 1").roundness;
hPad = tWidth * effect("H Padding")("Slider") * .01;
vPad = tHeight * effect("V Padding")("Slider") * .01;

box = [(hPad + cornerPad), vPad] + [txtBase[0], txtBase[1]];
[box[0], box[1]]


//Shape Layer/Contents/Rectangle 1/Position

//source layers 
src1 = thisComp.layer(index - 2);
src2 = thisComp.layer(index - 1);
// text size
tBox1 = src1.sourceRectAtTime();
tBox2 = src2.sourceRectAtTime();
// max width
tWidth = Math.max(tBox1.width, tBox2.width).toFixed(2);
lyrSpace = src2.position[1] + src1.position[1] - thisComp.height;
boxHeight = tBox1.height + tBox2.height;
refScale = thisComp.layer(index - 2).scale * .01;
x = tWidth / 2;
y = boxHeight / 2;
lineSp = effect("Leading Middle")("Slider")/2;
t = tBox1.top + lineSp;
// max width
maxWidth = Math.max(tBox1.width, tBox2.width).toFixed(2);
// set position
if (tBox1.width.toFixed(2) == maxWidth)
	l = tBox1.left;
if (tBox2.width.toFixed(2) == maxWidth)
	l = tBox2.left;
newBox = [(x + l) * refScale[0], (y + t) * refScale[1]];

[newBox[0], newBox[1]]


//Shape Layer/Transform/Position

thisComp.layer(index - 2).transform.position

You will need to add expression control Sliders to the Shape Layer and name them H Padding, V Padding, and Leading Middle. If you don't want to add sliders, you will have to type in values for the padding and leading where the sliders are shown.

That should get you started. I just pulled these expressions from one of the Animation Presets that I have that also include additional movement based on the layer in and out points so no keyframes have to be set to make a multi-line graphic animate into the frame and then move out.

I am working on a tutorial series explaining how sourceRectAtTime works in detail and simplifying the math so you can create your own automated animations using text and shape layers. 

One more note, the Paragraph Justification for both text layers must match. With a few more calculations using left and top values, you can mix and match paragraph justification. These expressions compensate for the baseline shift in the text layers. 

Just incase I made some typos in the expressions I have uploaded an AE 23 project file. Let me know if you need an earlier version.