Given that Paterson has stoutly denied ever looking at any CP/M code (and nobody has ever produced any contradicting evidence), it's clear that in his “quick and dirty operating system”, he took the CP/M user documentation and built something with somewhat the same architecture and user-visible behaviour, using his own code, and tailoring it to the somewhat more powerful 8086. OS designers have been doing this for decades: look at the documentation for Univac and RCA lookalikes for System/360 (available on bitsavers), and compare to DOS/360. Or that “hobby” system written by some guy named Torvalds...
CP/M has no tree shaped file system. MS-DOS borrows from Unix as much as from CP/M. For instance the ".." directory being parent (in spite of there being no such directory entry), navigated by a "cd" command, and command pipelines separated by | (in spite of there being no multitasking; all done via temporary files under the hood).
Later versions of MS-DOS had a "Xenix" API, inspired and named after Microsoft's Unix flavor.
The term "BIOS" comes from CP/M, referring to the low-level hardware-abstraction layer of CP/M (Basic Input/Output System), the machine-dependent part.
IBM PC family machines put the BIOS into the machine rather than OS image, allowing machine-language programs to take it for granted that they can invoke BIOS routines.