Even in the Regency, women usually are said to have a fortune rather than a dowry, which means money that belongs to them whether they get married or not. A dowry was specifically given upon marriage, and I'm sure in some cases fathers did give extra money, as part of the marriage settlements. I've only seen a couple of seasons of Downtown Abbey, but I seem to remember in the first season, when a duke was showing interest in Mary, her father assured him that although she would not inherit the Abbey, she would still receive a substantial dowry. Something like that.
