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According to Neil Gaiman's account,

Shadow's real name is Baldur Moon, which translates to the fact that the protagonist of American Gods is a god.

How is it possible that he does not realize that? Or does he?

Actually, there seem to be an apparently convincing argument that Shadow/Baldar is NOT Baldur the god, but merely has the same name. The full discussion can be seen here. The main points of the orignal write-up are:

First, it doesn't fit at all with the mythos established in the novel for Balder to be incarnated in human form. We see no other god that is embodied this way. It makes little sense for Balder to emerge now, since there's not exactly scores of people just starting to believe in him again. Besides that, the people that did believe in Balder for the most part believed he was dead, so he's just not going to be up running around in any form.

Second, while Shadow corresponds to Balder in many ways, there are many ways in which he doesn't. Shadow may be attractive, but he's not the Nordic vision of blond beauty that Balder was. Balder's wife died after him, not before. When Balder did die, it did not involve hanging himself in the manner of his father. Shadow's death does not involve the manipulation of a blind of otherwise handicapped brother, which I think is a pretty important part of the Balder myth. Shadow is a fighter, Balder wasn't.

Third, Shadow is a great human character, and it cheapens him to be seen as just another face for an eternal god. Gaiman's works tend to be built on normal, non-mystical characters encountering a world of magic. Richard Mayhew, Tristran Thorn, Rose Walker, that kid in Mr. Punch. Sometimes they turn out to have a mystical heritage, like Rose and Tristran, but they still tend to be very human. I see Shadow as very much in this mold, and probably the best of these characters so far. To make him a god strips him of his humanity, which weakens him in my opinion.

Shadow has many similarities to Christ as well as to Baldr: choosing to sacrifice himself, son of a god, being tied to the tree to die in way similar to crucifixion, returning from the dead, being resurrected by EASTER (which, especially after making the point that Eostre is not remembered except for Christianized Easter, is a very funny, loaded, and self-aware moment). Plus his appearance in "Monarch of the Glen" is to fill the same ritualistic role as Beowulf. This is clearly NOT an argument for Shadow being an actual reincarnation of Beowulf or Christ. ;-) (Indeed, Gaiman wrote a scene of Shadow, while on the tree, hallucinating/astral projecting into a conversation with Christ. It's read by Gaiman in the 10th Anniversary Audiorecording, and may have been included in the second printed edition, but was cut from the original publication because Gaiman says he wasn't sure it was part of the same book. Not for another reason, like wanting to keep it ambiguous that Shadow might be Christ.) Since no one seems to find that problematic, being clearly a thematic similarity, bringing together and reusing a lot of religious tradition/imagery/cross-cultural similarities in the same way that Christianity itself historically did, I don't find it problematic for Shadow also to have similarities with Baldr in a thematic way, without it needing to be literal. Even in a work where a lot of gods do literally show up.

Also just a side note balder is attached to many old weapons and armors. Try to remember that he has the protection of all but one thing, in a pantheon of warrior gods. So it could be argued his protection wouldn't be wasted on drunken games. Though this is how he dies it would seem Odins second son would have great importance to him. Why else do you think Odin wants him. If he were human he coulda cared less. Shadow was his son ,and after how many he losses already I can see him not wanting the favourite to die. I think shadow is reincarnated version of his old self . That's y he has no idea because memories being lost perhaps. As a New born God sort of speak. The youngest sister states that he once had the protection of the sun. But she can give him the moon. Also thought since his personality is the exact opposite of balder , but not at first. Prison and a woman who had no faith brings his very happy self into the angry antagonist we see.