If it is that particular image, then try changing to Object Aware — I think the default is Color Aware
Still not perfect, but that is not an easy selection.
You can improve it by increasing Smoothing and contrast to fix the jagged edgea, and Decontaminate Colours to remove the green tinge. It really depends how you intend using the selected image, and what you would be laying it against.
If your issue is with all images, then show us some more examples and maybe some more information.
OK, looking at your post again it is specifically the Background Removal tool. I would say that is not the best subject to use such a broad based tool, but my system still did a better job than yours.
If you click on View More you get the option to Refine with Select & Mask
If you click on that and paint of the problem areas with the Refine Edge tool it cleans it up heaps. But better still to select something like the Quick Select or Object tools and click on Select Subject.
Use the drop down and set it to Cloud (detailed results) and I suspect you'll get much better results.
Clean it up with Select and Mask and you'll be golden.