Why the washing cup has two handles
We couldn't put it better than Rabbi Eliezer Posner on Chabad.org:
"After the first hand is washed, it is clean and pure. The unwashed hand, however, is not. If the two hands touch after the first hand was washed, it is necessary to rewash the first one. We use a two-handled cup to make the process simpler, making it easier to avoid the hands touching each other.
You can use any cup to wash your hands, just be careful that the two hands don't come in contact with each other after the first one is washed."
"After the first hand is washed, it is clean and pure. The unwashed hand, however, is not. If the two hands touch after the first hand was washed, it is necessary to rewash the first one. We use a two-handled cup to make the process simpler, making it easier to avoid the hands touching each other.
You can use any cup to wash your hands, just be careful that the two hands don't come in contact with each other after the first one is washed."