The commands are different because they were developed pretty much in parallel, without interaction. DOS commands are generally derivative of QDOS, which itself was derivative of CP/M, Gary Kildall's Control Program for Microcomputers, which was pretty much the first operating system for Intel microprocessors, originally intended for Intel's 8080 CPU, and written in large part to enable PL/M to control a floppy disk drive. This all happened in the early 1970s.
Linux is largely derivative of Unix, with many of the commands having identical names performing identical (or very, very, similar functions). Unix was being developed starting in 1969, and so much of the initial command naming happened in parallel with CP/M.
CP/M was aimed at personal computers, while Unix was aimed at minicomputers, generally in corporate labs, or university research centers, and there was really no collaboration between the two efforts.
Now it's largely kept the way it is to ensure differentiation, with neither side willing to admit the other's naming convention is superior.
Both systems allow command aliases, though.