Law Degrees in the US are not limited by some quota beside the size of the classes and the number of schools accredited. The real barrier is the Bar Exam, the ultimate decider if you are allowed to practice law, every state has there own admission process, there is some reciprocity and I think the Federal level has it's own separate Bar standards.

Not in the same way. IANAL and IANAD but I think:

IN LAW {bar exams + prestigious law firms} ~ IN MEDICINE {board exams + specialty board residency placements}

But, there is no need to work for a law firm prestigious or otherwise to practice law, you just pass the Bar.

Not really. I suspect this is because unlike medicine, lawyers as a collective organization don't benefit as much from an artificial scarcity. MD isn't really a common path to anything other than a career as a doctor, but JD has other practical career paths.

It also differentiates the bar vs. board exams. You can't practice medicine without certification, but there are plenty of lawyers doing law without a bar association, it only constrains you from doing certain types of work.