Uhmm, excuse me, ...
... as someone who does have a JD, and an LLM, I have to say that calling someone with a JD "doctor" is like conflating a bachelor's degree with a PhD. The JD was the in-your-face replacement for the old - honest - law degree, the LLB, aka, Bachelor of Laws. That's why the LLM, aka Master of Laws, comes after, not before, the JD. The law school equivalent of the PhD - the "doctoral" degree - is the JSD, the
Doctor of Juridical Science.
There is no real equivalent to the JD in the normal college > grad school > doctorate progression. The closest one could come would be to analogize it to an associate degree - the degree that's frequently received from two-year colleges; that is, the JD is sort of like an associate master's degree - higher up the chain than a bachelor's degree, but still less than a master's degree.
In point of fact, calling a lawyer "doctor" on account of his having a JD is even more facetious than calling a lawyer "esquire" - and doubly so if the lawyer calls him or her self that.