Well, close. Linux is a free Unix "work-alike" in most respects, but it is definitely NOT Unix. It was specifically written to NOT be Unix, but rather to function mostly like it.
More to the point, the main difference between Linux and Unix is the kernel -- the innermost part of the operating system. They are completely different, but those differences are hidden below the surface of the command and application programs.
Both Linux and Unix use the GNU suite of system commands, and those commands are nearly identical in the two OSes in all respects. Linux and Unix also add their own OS-specific commands to the GNU suite.
The good news is, real Unix is free also, just like Linux. FreeBSD, NetBSD, and others are wonderful operating systems, if you prefer the great stability and security of Unix, and are willing to put up with a limited availability of commercial application software.
In most cases of "I want Unix", Linux is a much better choice, because you get a virtually identical experience from the commandline, and a much richer experience on the desktop.
I know all that you wrote. I have been working with Unix since the early 80s. I am fully aware of what you pointed out.
I just meant that the distinction between Linux and Unix nowadays is so minimal that is it meaningless, when discussing functionality as a desktop OS to an end user.
In the server world, there is a distinction, with difference. But we were talking Windows desktop replacements here, not server OSes.
(As an aside, I am not a rabid Linux supporter. I prefer the BSDs: NetBSD, OpenBSD or FreeBSD. Much more complete, coherent systems than your typical Linux distribution.)