This definition of the noun "cast" may not be considered archaic, but it's very unusual in modern English. In this case it means, "characteristic quality". It's rare enough that it's the second part of the tenth definition of the word as a noun. It basically means this person's basic life force was romance and seriousness. https://www.merriam-webster.com/dictionary/cast The female being talked about in this passage is "all life and gladness" which simply means that her personality was very lively, and happy. I note that this is from Washington Irving, who is considered one of the very first uniquely American authors, and who wrote in the early part of the 1800s. Here in the USA, students read Washington Irving during the first part of their American Literature course. It is written in 200-year-old English, so you may come across old-fashioned words and phrases. He is best known and beloved for the stories of The Headless Horseman (a horse-riding ghost who terrifies a small town), and Rip Van Winkle (a lazy drunk who falls asleep for 20 years). These have become very famous folk tales and almost all Americans will have heard the stories.

In this sentence 'cast' is a noun. If you made a metal object from a mould, the object so produced would be called a cast. They are categorising him as belonging to a group of people who are romantic and serious - they are all out of the same mould - of the same cast. The writer of the sentence could have used the word 'mould' as an alternative. She is the opposite of serious - she is bright and lively - we might say that she is the life and soul of the party!