The Xbox was Microsofts first console. The GC was Nintendos fourth.

True, but Nintendo's first console sold 60m and Sony's first sold 100m, with both winning their respective generations. 

A console being a first doesn't in itself mean low sales.

And with as dominant as PS2 was Sony still got scared of Halo and made their "Halo Killer" in Killzone and by PS3 made a bunch of FPS franchises hoping to get that Halo like success. Then there is Xbox Live which made a huge mark for online gaming. 

1. Xbox still outsold the Gamecube.
2. Xbox 360 was a sequel console just carrying on the core ideas from Xbox. Gamecube's ideas on the other hand were completely abandoned. In fact for better or worse, many of Xbox's ideas changed gaming as a whole.
3. Xbox had to take on Nintendo's 4th console and Sony's 2nd console after a dominating 1st console. Gamecube had to take on Sony's 2nd console after a dominating 1st console and MS's first console.

Besides selling better than the GameCube, it also was just taken much more seriously as a PlayStation competitor in the public discourse. Especially because of Halo, which was undeniably one of the biggest new IPs introduced in the 6th generation.

Another point was that it was only Microsofts primary system for 4 years, while the Gamecube was Nintendo's primary system for 5. The Xbox still had some momentum when it's successor released, while the Gamecube was completely dead.

All that being said, I am not really sure I'd consider the Xbox a successful console though, it wasn't a failure either, just somewhere in the middle. It was a strong stepping stone towards the 360, but I don't think the sales justify calling it a success, considering what Microsoft's goal was. For other companies, 24 million systems would be a huge success, for MS, not really.

They might have sold similar numbers, but the paths they took to getting to that number were different - the Xbox started out as a slow burner, but by the latter stages of the generation it was actually doing pretty well, even if nowhere near as good as the PS2. By contrast, the GameCube sold well for about 18 months and then utterly tanked. Not to mention that the Xbox outsold the GameCube despite selling terribly in most Asian countries.

Plus, while it obviously wasn't anywhere near the top-selling system for that generation, the Xbox did have a pretty big impact on gaming culture, introducing us to the likes of Halo and making people realize just how useful a full online ecosystem was for a time. By contrast, the GameCube, rightly or wrongly, was pretty much seen as just that system that you and you buddies got out whenever you wanted to play Super Smash Bros. Melee.

I think a part of it that hadn't occurred to me before, now that I think about it, was that as an English speaker, a large amount of gaming interactions are with Americans, where Xbox overperformed relative to the rest of the world.

It's like how, as a teen, I knew more people with an Xbox than a PS2, whereas globally PS2 sold like 6 times as much as Xbox, so my sense of how well it did was skewed and I have quite surprised to learn in later years that it only had like 10% market share.

Gamecube did better in Japan for example, so among Japanese gamers there's probably less of a sense of the Xbox having been a success.

That's pretty much my theory as well, combination of its influence on following consoles, Halo's cultural status, and the fact Xbox did better in the countries that make up the population of most English-speaking sites, despite relatively low sales.

Because the gamecube sold way less than the NES, SNES and N64. The GC had a benchmark and it was way under.

The xbox got its foot in the door, expectations were just different.  

I was there and view them both as failures. But maybe other people view the Xbox as being successful because its features went on to influence the industry whereas the Gamecube was sorta forgotten. That's all I can think of.