How to automatically size an image to fit into a selection or a shape easily?

You could use a smart object… and replace content…

Are you holding down the Option key to resize from the center?

The way I do this in scripts is: I resize the image layer so it will fill the bounds of the selection. Then I align image center to the selection center. Then I mask the image layer with the active selection.  If the selection is a rectangle the results will look like a centered aspect ratio corp of the resized image. If the image had the same aspect ratio as the selection the will be no image content masked off. If the aspect ratios were not the same. then some image content is masked off on the top and bottom or on the left and right sides. The selection  actually can be any shape it doe not matter what shape the selection is the process works.

IMO Adobe Frame Tool has issues I removed it from my tool bar and disabled its shortcut.  

If you have a shape you can Ctrl+Click on the Shape layer to set a active selection of the shape.  As I stated above the selection can be any shape. And when added to the image layer as a layer mask will control what areas of the resized image will be visible.

Adobe Frame tool does not support any possible shape.

Frame tool can shrink image to fit inside but instead of pasting you will need to place image. Perhaps that can work for you? Otherwise I know for Variables > Pixel Replacement > Fit which can automate that process but that is also connected with importing new image and you must have area in which you want to fit present as pixel values on layer not as selection.