In this episode Dr. Mike & Daniel are encouraging Matthew to charge people for the law advice & services he provides. But Matthew says it isn't the same as Dr. Mike being a doctor because she passed a test to become a lawyer, but that "there's not a test to become a lawyer." My curiosity got the best of me and I decided to research the issue. I found that the College of William and Mary established the first chair of law in the US in 1779 and that the first bachelor of law was awarded there in 1793. Law was taught at the college until the Civil War when the school closed; it re-opened in 1920.
In addition, the University of Georgia School of Law opened just before the Civil War. So I am sure there was a law school operating in the U.S. in the early 1870's, which is when I estimate this episode took place since it was sometime in 1868 when the series started and this episode is near the end of the 6th season.