The key here is to recognize what the numbers (1), (2), (3), and (4) represent. Are they just names (nominal, e.g. Vanilla, Chocolate, Strawberry)? Are they ranks (Ordinal? In what order do you prefer Vanilla, Chocolate, Strawberry)? Or do they represent a precise interval (On a scale of 1-10, how many units of happiness does each flavor give you?) Or do they represent a precise ratio? (Does a rating of 2 mean that you prefer Vanilla twice as much as a flavor you rate as 1 or half as much as a flavor that receives a 4).
Applied to party identification, ratio does not make sense because a democrat is neither twice a republican nor half a "not registered voter". And interval fails as well because in what sense is a democrat 1 unit than a republican or 2 units less than an independent. Similarly, ordinal makes no sense because the question is not asking for a ranking among the 4 numbers. Thus, the data must be nominal, which makes sense because all the numbers do is "code" or substitute for the respective names or categories.
The explanation that this is nominal data because the numbers are substitutes for the names or categories is probably a sufficient answer, but I hope the longer answer can assist you with the other problems that are typically assigned giving examples of all 4 categories.