After doing a factor analysis, I decide to use that factor to test a hypothesis. However, I am not sure what data type of the Factor score the is (i.e., nominal , scale, etc.). Can I do a t-test or ANOVA as usual?
Factor scores are interval (and therefore scale, because "scale level" = "interval or ratio level").
Obviously, they are not nominal. Are they ordinal? Ordinal level means that the observed variable is related to the true, underlying feature in some (unknown) monotonic, rather than linear, way. According to linear factor analysis theory, variables are linear functions of factors; also, according to linear factor analysis practice, factor scores are computed as linear functions of variables. Here is no room for nonlinearity to emerge. Thus, if you treat your analysis variables as interval the factors you get are interval logically and automatically. If you theat the variables as ordinal you should carry that stance over to factor analysis, for example by doing variable transformation via categorical PCA (CATPCA) in place of or prior to linear factor analysis. For, if you worry that the variables are ordinal but nevertheless proceed to linear factor analysis with them unaltered you actually take them to be interval anyway.
Factor scores are certainly continuous. Whether that makes them interval is an interesting question, but I think they can be treated as interval for most purposes.
It's an interesting question because Steven's classification says that something is interval if intervals are the same. Is the difference between a factor score of 1 and 2 the same as between 0 and 1? Well, it depends on what your definition of "same" is.