Is alkene reaction with hydrogen to form an alkane an addition reaction? I have seen in some textbooks that such a reaction is listed under reduction of alkenes.

I understand that the double bond is getting reduced to single bond and one hydrogen atom is added to each of the carbons. 

Should I classify this reaction as an addition reaction or a reduction reaction?
Yes, the hydrogenation reaction of alkenes is typically classified as an addition reaction. See here and there, for example.