No. AT&T created Unix but was unable to market it due to a previous antitrust action. So they gave it away. (They required a license signature (I've signed that! :-)), but did not charge and were very lenient.)

The UCBerkeley Computer Science Research Group (if I've got my acronyms right) was one of the recipients and went on to add on many, many features and releasing the result under the BSD license.

Many people built companies like Sun around selling BSD Unix, including many alumni of UCB.

Then AT&T got out from under the consent agreement and began selling its own Unix, System V. By this time, Unix was a major player in the workstation market (a market that has largely disappeared as PCs got more powerful).

By, say, the mid 1980s, there were many, many companies selling many, many varieties of Unix, all descended from the original Unix via BSD or via BSD+System V. Most of them had some unique, valuable features (Irix's graphics, AIX's LVM and journaling file system, etc.) and all of them had modifications to lock customers into their version. This is where the POSIX standards come from (second only to ecmascript market manipulation goofiness), and why things like Autoconf/automake and the much-loved imake (not really) exist. There was much in-fighting; Sun vs everybody else, everybody else vs. IBM, etc.

Then two things happened: PCs got more powerful and began eating into the bottom of the workstation market (PCs ran DOS+Windows, which was and arguably still is, technically inferior to multi-user-by-design systems like Unix[1].) And PCs got more powerful and began to be able to run more advanced OSs (think "memory management").

At this point, the Unix world began to conflict with the Windows world. Unix was technically superior, Windows had more public and developer mind-share. But the Unix world was still more interested in fighting each other and stapled all of their arms and legs to that particular tree.

The end result was that Windows became and remains the most-used operating system[2]. All (almost) of the commercial Unixs died (almost; there's still some animated corpses around)[3]. The two counter-examples are MacOS, which is completely locked to Apple hardware, and Linux.

Linux is the interesting case. Windows and commercial Unix all had a 15- to 20-year head start. But Linux achieved (mostly) feature parity quickly and did not break down into multiple, competing streams. Both of those are due to the GPL; you can fork GPL software all you want, but you cannot add a feature to a fork and expect it not to be back-ported into the original if it's useful. You also have a very hard time locking users into your fork.

The bottom line is that Microsoft won the Unix wars, because the Unix licenses allowed companies to take Unix proprietary.

[1] Modern Windows is kinda-sorta based on VMS, another workstation OS, but not really and then they walked that back, and so on....

[2] I don't really consider Android or iOS to be general-purpose OSs. And they're both rather their own little islands, no matter how much the underlying tech shares with the rest of the universe.

[3] The Free/Net/Open/DragonFly BSDs are, I'm sorry to say, noise. And did you notice that I had to mention four of them?