Yes, that's an interesting question. The short answer is that most variants only have two alleles, and even though you can be heterozygous at a position in two different ways, both of these result in the same genotype.
We can have A, T, C, or G at a position.
Since there are two chromosomes, we could have many genotypes: AT, AC, AG, CG, GC, etc. However, most of the base pairs in all mammal species are the same. Only a small fraction have even one alternate allele. That is, most positions just have one letter in them in all humans (and animals), and there isn't any other letter anywhere in the population to make an alternate genotype. However, sometimes there is this extra letter available in the population. This is what we call a "variant." Since this is somewhat rare for there to be a variant at a position, it is even rarer that there are more than one variants at a single position. GWASs therefore usually only consider the case where there is one variant. That is, there are two possibilities for the DNA "letter" at that position.
Let's imagine these are C and T. The possible genotypes are C C, C T, T C, and T T.
T C and C T are the same genotype, because they both have 1 C and 1 T.
This means that there are three unique genotypes for the case where there are two variants. This is where the 0, 1, 2 coding comes from. 1 refers to the heterozygotic position (C T or T C, in our example).
In reality, it is possible to have more than two variants. These are called multiallelic sites, and they are not uncommon, but they are rare enough to not be worth the trouble of including in GWAS, so they are either excluded or split up into multiple alleles, each with two variants. There is some work on modeling these more effectively.