OK, I'm going to delurk for this one.
Sometime in 1998, I attended a talk Linus gave in Silicon Valley. It was, I gather, his usual thing: he just answered questions, he didn't really have a speech. I don't remember how the subject came up, exactly, but he started talking about his goals for Linux.
He said, approximately, "I'm going to take over the world." And the audience laughed. "You think I'm kidding, I know you do. But I'm not. I'm going to take over the world." And I will tell you this: he meant it. There was a sharp look in his eye; an uber-geek at the height of his technical power, knowing that what he was doing was going to change everything. He knew it sounded ridiculous, but he was being absolutely honest, not joking in the slightest. He intended to dominate everything with his operating system.
I believe that, at the time, he thought of "the world" as being "the desktop". That was the center of computing back then; servers were rare, and the Internet was pretty young, but everyone had desktops. Linux was never really intended as anything except a desktop OS, at least in the beginning. But that early code was so amazingly reliable, compared to what Microsoft was offering, and so incredibly cheap, compared to what Sun was offering, that it ended up pressed into server duty. It was the accidental server; it fell sideways into that role, and ended up being one of the best solutions on offer, simply by the virtue of a clean design and good code.
So, here we are, fifteen years later. And, you know what? Linus *did* take over the world in most respects. But he didn't do it how I believe he expected to; I'm pretty sure he thought he would break the Windows monopoly directly, and take over everyone's home computers. But Microsoft has largely managed to defend itself there. Rather, Linux turned into an ecosystem, and it went around Microsoft, growing into many other markets. You still don't see it on the desktop, but it is absolutely ubiquitous everywhere else. Chances are quite good that any reasonably prosperous household in the First World is running Linux somewhere, possibly in multiple devices. They may even be running more Linux than they are Windows, without even realizing it. And if you use the Internet, you're talking, at least somewhat, with Linux machines. It's in phones, it's in tablets, it's in routers, it's in servers, it's in supercomputers. It's everywhere except the desktop.
Honestly, I think it would have continued to make inroads even there, but the GNOME team and Canonical went nuts chasing tablets, burning their existing users and giving them a horrible desktop product, trying for imaginary tablet customers that never materialized in any significant way. Ubuntu Linux, in 2010, was an outstanding desktop OS, and three years later, it still hasn't regained that usability, discoverability, and just general functionality again. If those desktop teams hadn't lost their collective minds, I think Linux would be doing well, even in the center of Microsoft's power.
That, however, didn't happen. None of the assaults on the Microsoft desktop fortress have ever been successful. Microsoft still rules there. But Linux is either a major player or completely dominant in every other computing market that's developed in the last fifteen years. And the advent of cheap-as-potato-chips computers, like the Raspberry Pi, will only increase that effect, as new markets arise, and Linux is adapted to fit them.
So, no, Linux has not won in the way that Linus originally intended. That battle was lost, decisively. But he and the kernel devs have thoroughly won the larger war.