Command prompt in Windows and Linux -- What is their language called?

In Windows, when you open a command prompt, almost always the program that is running is cmd.exe. It's an enhanced NT derivative of the old DOS-based command.com, which is in turn has similarites with the even older CP/M CCP command interpreter.

In Linux, as @Griffin mentions, usually you are running bash in some sort of terminal emulator if you are using it in a graphical environment. (which can vary widely depending on your distro and desktop environment). This stands for "Bourne Again SHell" because it's an extended version of an earlier "Bourne" shell (sh) that's been standard on Unix for a long time. There's other "alternative" shells in common use such as csh, ksh, and tcsh.

The Windows cmd.exe can be considered a shell as well (as can explorer.exe).

Both the Windows command prompt and all Unix shells have several "built-ins", or commands that are handled entirely within the shell. This includes some basic commands (such as cd) and many conditional commands and operators that control script flow if a batch file of commands are executed.

However, in both Linux and Windows, the great majority of "commands" that do useful things are in fact external programs that are "called" by the the shell. And that is the primary purpose of a shell, to enable an operator to start programs with specifying arguments for those programs. It's not really strictly a programming language, it's just a framework for launching programs.

It's not a "language" really. It's simply the Command-Line Interface (CLI) for that particular operating system.

The commands and syntax are chosen and defined by the operating system creators.

There are various scripting languages (some more popular that others, depending on the operating system, etc.) that are commonly used with in conjunction with a CLI to perform batches of tasks.

BTW the term "Command Prompt" refers to the actual bit of text that signifies where you are to enter your next command in the CLI. (ie: C:\> or #, etc.).

Windows uses batch. The most popular language in Linux is bash, but there are alternatives.

MS Windows Shell Command Language?

In comparison to

https://pubs.opengroup.org/onlinepubs/9699919799/utilities/V3_chap02.html

In Linux, or any other Unix-based system, they call it shell scripting.