How can I fit this bottle holder to this bike? I want to fit this bottle holder on my bike but the attachment seems too thin for the frame tubes. I believe this is a universal bottle holder and my bike isn’t of unusual size or tube girth. Am I missing something here or this just won’t fit?

P. S. Unlike my mountain bike, this bike (Claud Butler Legend) does not have pre-drilled holes for accessories on the frame.
That does not appear to be a attachment for mounting a bottle cage to a frame tube. The two holes in the mount appear to match the two holes in the upper plate of the cage, which orients the mount incorrectly for clamping a frame tube. My guess is that it's for mounting to handlebars - which have a standard diameter, unlike frame tubes.

mounts for frame tubes do exist. They typically use two adjustable straps of some kind to accommodate different size frame tubes. You want something with a beefy strap that can be tightened sufficiently to stop the cage rotating on the frame tube.