I don't think there really is a great askhistorians worthy answer for this so i'll give my answer a go which has stuff drawn from other census information: it's not a race. It's an ethnicity. The classical racial divisions don't treat the levant/MENA as a separate racial category.

Wikipedia tells me there are 1.7 million arabs in the US today but 60% of them are Christian (even with larger influxes of non Christian arabs) and that's part of the deal. Races come and go on the Census based on demographic (and some ideological) pressure. Arabs never really migrated in a critical mass to the united states and many that did (say "lebanese" Christians at the turn of the 20th century) weren't culturally strange or threatening enough to merit a different racial classification for othering purposes. If Muslims from Syria and Palestine had been a major group of immigrants the race question would have been more pertinent (but that's speculative). Mexican/Hispanic got asked because it became a pertinent question to ask and thus is lumped in with races. Essentially with the decline of racial pseudo science and the rise of some more modern social movements the census may morph again and label Arabs as a race-ethnicity along with some other groups but the key to understanding why it does or does not lies in general societal ideas more than an exact science.

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