Your image is not perfect square while handles are not following shape but rather 90 degree angle in each corner. Everything is OK, thats how things work in Photoshop. If you drag corner handles beware that you will distort image in attempt to make it square. Have you tried to mask image or to clip it to square shape?
I know the image isn't a perfect square. My goal is to free transform it into a square but when I select the image, the selection doesn't adjust to the actual shape of the image, but creates a square around the image as you can see in the provided example. Typically, the selection should adjust to the actual shape of whatever the layer is, and then from that shape you can Free Transfrom by clicking and dragging the tiny squares in the corners of the shape. While it may cause distortion, I don't believe that adjusting this image to a square will distort it badly as a square shape isn't too far removed from the shape that it is currently.
So I'm wondering why the selection doesn't put the tiny squares to the actual shape of the layer but rather creates a square, which means those tiny squares aren't actually touching the shape of the layer?
I think what you need to do is to firstly select View > Show > Grid. Make your selection, then select Ctrl+T to show the transform handles, Right click on the image and select 'Skew'. Click and drag the necessary corners to pull your image out to the lines - not diagonally, vertically then horizontally. Click the green tick, then Crop (Image > Crop). Hope this helps.