I'm pretty sure C/PM 86 was already a thing. I believe there was even Concurrent CP/M 86, which let you run up to 4 (or more?) virtual sessions concurrently. Light years ahead of MS-DOS but nobody wanted it, so DRI dumbed it down and added MS-DOS compatibility and re-released as DR-DOS (still superior to MS-DOS though).
CP/M-86 was definitely a thing, but CP/M-386 was not. By the time the 386 existed, DRI had dropped the CP/M name in favour of Concurrent DOS - though they had MS-DOS compatibility in Concurrent CP/M - and, as you say, eventually renamed it to DR-DOS.
Much as I loved DR-DOS, it did have occasional compatibility hiccups.
Wasn't that abbreviation from "pipe"? As in - this is a pipe that you can use to connect terminal to a file, file to a printer... and yes, file to a file (that's cp/copy).