This link shows a sample enumeration page. http://c.ancestry.com/pdf/trees/charts/1850.pdf?cj=1&o_xid=0002489095&o_lid=0002489095
There is no "ethnic identity," there is no "racial identity." You got the choice of White, Black, or Mulatto. Which one do you think the Irish would have chosen?
Perhaps you confuse the bias against "non WASPs" with an idea that Irish were not considered "White"?
By the 1870 census the categories were: White, Black, Mulatto, Chinese, Indian. Same in 1880.
These early usages conform with the scientific thinking of the time that the three great races were Caucasoid, Negroid and Mongoloid.
Hope this helps.