The word ‘white’ means a lot of different things. Those groups may not have been considered white socially but they were always legally white for the purposes of immigration, not being enslaved, not being subjected to Jim Crow.
As a result, populations poorly represented in 1890 were prevented from immigrating in proportionate numbers—especially affecting Italians, Greeks and Eastern European Jews, as well as Poles and other Slavs.[1][3][4] According to the US Department of State's Office of the Historian, the purpose of the act was "to preserve the ideal of U.S. homogeneity."