Did people actually drink from these cups?
Yes, it was most likely used! Although objects produced by Tiffany Studios were beautiful, they were designed to be functional. The company was established at a time when the Aesthetic Movement was the dominant decorative style in the US. Aesthetic Movement designers like Louis Comfort Tiffany believed that things should be as beautiful and artistic as possible.
This case of Tiffany objects is like THE Tiffany; the one who makes the silver jewelry today?
Not quite, but close! The glass company was founded by Louis Comfort Tiffany, son of Charles Lewis Tiffany who founded the silver company that has its headquarters on 5th Avenue today.