How is sugar cane and sugar beet processed into packets?

You first wash the cane/sugar beet. Then you crush them to get as much juice out as possible. This juice contains a lot of sugar, but at this stage there is also a lot of water (hence why it is juice...

Both can be made into sugar cubes, as both beets and cane are high in sucrose.

Assuming that you are asking about cubes made from white sugar, then the difference between sugar made from beet or cane is nonexistent. Both are pure sucrose; they don't really have a distinguishing...

Sugar beets can be turned into beet sugar

I thi;nk Kroger is more likely to use beets, If the store were selling cane sugar it would surely be on the label as that is the most desired sugar.