What they are drinking from is saucers- yes, it is odd! The idea was that you poured your hot drink into the saucer to cool it a little before downing it.

Teacups with handles did exist in those days, as many museum specimens will bear out.. However, if you look at the scenes in the movie from the farmhouse, and if you look at the most recent S&S movie, you'll see them using teacups the same way as we do.
I would suggest that the use of saucers in that particular scene indicates that the Inn is rather an old-fashioned place, and whilst the gentry might be used to handled teacups, the landlord did not expect all his customers to know what to do with them!

More info? Here's two websites that have interesting articles and some pictures that show both handled and non-handled tea and coffee cups.
http://www.cityofdunedin.com/city/?page=feat_teapot

http://www.lady.co.uk/articles/0117artA.cfm?framed=y

Question.... At Mrs Bennet's dinner party at the end of P&P Jane pours the tea and Lizzy the coffee . Would they have used different styles of cups, do you think?