It is and isn't. There is a god called Lodurr that has very little information about him. He's referred to being Odin's brother (or possibly his friend) at one point in one writing. "Lodurr" (meaning the name, written out) looks very similar to Loki in norse, so some scholar's believe that Lodurr actually refers to Loki. And this is especailly confusing because there there are indeed several mythical beings whose names are identical or similar to Loki, like Loki of Utgard and Logi. The research I've looked up says the Lodurr-Loki connection is superficial as there is little connecting them and the few mentions we have of Lodurr have him isolated and acting uncharacteristic of Loki, so it's probably not how most people saw Loki, but at the same time it's not like this stuff doesn't happen. Odin, for example, is both an Aesir and a Vanir. His Vanir name is Odr. This is theorized that at some point, the Aesir and Vanir were just one group of gods and got split. When they did, Odin was the Aesir aspect while Odr was the Vanir, but they are actually the same god. So it's unlikely, but possible, and Gaiman probably just went with it.
But listen, I feel that people misunderstand one extremely simple point about mythology: There is no canon. The vast majority of mythology was told orally. Ever play a game of telephone? Imagine hundreds of thousands of people telling a bunch of different stories across an entire landscape for hundreds of years. Writing things down came way, WAY after most myths have been told, and even writers contradicted each other over the years.
All that it really takes to make Loki Odin's brother is some random guy deciding he wants to tell a story where Loki is Odin's brother. In this case, it seems this particular invention didn't get as popular as Loki being a son of giants that somehow gets to hang around Asgard, but functionally speaking, that's how all myths are told. To put it simply, if you're the kind of guy that gets hung up on continuity errors, mythology is going to drive you crazy.
Edit: Keep in mind that Gaiman contradicts himself in his own book, in the very next section even. "Players" is where he mentions that Loki is Odin's blood brother and he also says that Loki's mother is Laufrey and his father "is said to be" Farbauti. But in the very next chapter that explains how the world was created, it's mentioned that Odin is the son of Bestia and Bor, and his brothers there are Ve and Vili. My best guess is that perhaps Gaiman meant blood brother in a metaphorical sense. Like maybe they had fought together somewhere, becoming brothers in arms except they decided to call it blood brothers because of all the blood they spilled together? Or maybe Bestia, who is mentioned to be a giantess is another name for Laufrey? That's about all I can speculate on. It's more likely that Gaiman either didn't notice, didn't care, or maybe even wanted a few continuity errors to make it feel like he's retelling myths he heard rather than he made up (which he...basically is.). Mythology is where continuity dies crying helplessly, and you just need to own that.
Edit 2: Actually, the fact that he specifically wrote out that Loki's father "is said to be Farbauti" is probably his way of covering his ass about it. So, in his version, Loki's mother is definitely Laufrey, but Loki's father is said to be Farbauti, but it's actually apparently Bor.
Edit 3: You know, one aspect I find interesting about mythology is that, because it's nature is that there is no canon, you could argue that it never stopped, so modern retellings of the stories aren't just retellings, but actual additions to mythology itself. Even stuff like Wonder Woman or God of War, or Marvel's Thor is just yet another addition to their respective collections.