The answer is yes, a JD is a doctorate.
The JD was established as a professional doctorate in the United States in 1870, just a few years after the first research doctorate (PhD) was established. According to wikipedia, professional doctorates are awarded in certain fields where scholarly research is closely aligned with a particular profession, such as law, medicine, or psychology. (source)
Does this mean you are allowed to call yourself Doctor so-and-so after you graduate law school? Yes, as you are free to do anything that isn’t illegal, but expect to rapidly lose friends. Note also that calling yourself a doctor and doing anything outside your training can expose you to fraud charges.
The lay understanding of the word “doctor” is that it applies only to doctors of medicine and psychiatry, and to a much lesser extent to doctors of research who are professors. In the USA, the general rule is that unless you have patients, you are probably better off not calling yourself a doctor.
Some might even quibble with you if you argue that a JD should be considered the full equivalent of a PhD. For example, Elie Mystal, a blogger at Above the Law takes exception to the ABA’s resolution that JD’s working in academia be treated the same as anyone holding a PhD in this piece.
The debate over the relative merits of different post-baccalaureate programs can get a little heated, but just remember that the debate changes nothing about the innate value of a law degree no matter what it is called. The legal profession, titles aside, is an ancient and venerable one and is likely to remain so.