Conceptually, nominal/ordinal measurements are NOT discrete.
Note that nominal and ordinal measurements are classified under categorical data. That is, they are born as characters, but they can be coded into numerical values. On the other hand, discrete data are quantitative, which means they are born as numerical values. Therefore, nominal/ordinal measurements are conceptually different from discrete measurements.
Having said that, the way we analyze ordinal data may sometimes be similar to that for discrete data, provided we have enough categorical levels. For example, when we measure preference on an 11-point Likert scale, i.e., 0, 1, 2, ..., 10, we may analyze the data as if they were discrete.
However, nominal data can never be analyzed as if they were discrete. Why? Note that we may always rank discrete data but not nominal data.
For certain, interval/ratio measurements may not necessarily be continuous. For example, the number of hospital visit is a ratio measurement, but it is discrete rather than continuous.