Are the Command Prompt and MS-DOS the same thing?
This was true once, but it isn't anymore.
From MS-DOS # Windows command-line interface - Wikipedia:
All versions of Microsoft Windows have had an MS-DOS like command-line interface (CLI). This could run many DOS and variously Win32, OS/2 1.x and Posix command line utilities in the same command-line session, allowing piping between commands. The user interface, and the icon up to Windows 2000, followed the native MS-DOS interface.
Consumer Windows (up to 3.11, Win9x, WinME) ran as a Graphical User Interface (GUI) running on top of MS-DOS. With Windows 95, 98, and ME the MS-DOS part was integrated, treating both operating systems as a complete package. The command line accessed the DOS command line (usually command.com), through a Windows module (winoldap.mod).
A new line of Windows, (Windows NT), boot through a kernel whose sole purpose is to load Windows. One can not run Win32 applications in the loader system in the manner that OS/2, UNIX or Consumer Windows can launch character mode sessions.
So no, in every Windows from the NT family (e.g., XP, Vista, 7, 8), the command prompt and MS-DOS are visually similar, but quite different.
They are different things - The Command Prompt is Not MS-DOS - but as far as the user is concerned they could be the same thing as they do the same things.
So it depends on your point of view. From a technical point of view your friend it correct, but from a user perspective you are correct (sort of as there are differences that an expert would spot).
No.
(Unless your definition of equality does not extend past »It is a text interface and I can run programs from it.«)
What is run when you click Command Prompt in the Start Menu is the Windows Command Processor, a.k.a. cmd.exe. Its built-in commands and scripting syntax (including many quirks) are based on the ancient command.com from CP/M and later MS-DOS, but apart from that they are completely separate things. Also, command.com is a 16-bit program while cmd.exe is a native Windows console application.
Things were different in Windows 95, 98 and ME where command.com would be run in a MS-DOS VM with Windows acting as the hypervisor (yes, they had that sort of thing at the time already). There you had an entire virtual machine running DOS. But on Windows NT, 2000, XP, Vista and 7 – no. DOS only lives on there in ntvdm.exe which is the NT Virtual DOS Machine which is just a thin emulation layer capturing calls that the CPU cannot execute directly (which is why it works faster but worse than DOSBox).
In any case, even command.com was just a shell for DOS. It wasn't the operating system.
Inside, I actually cringe each time I see people referring to a window with gray-on-black text as MS-DOS. In the vast majority of cases they don't actually know what they're referring to.