Measuring the level of anonymity is not an easy task. It can be easier in some exceptional cases, but that is not true in general. For example in an anonymized database, we could measure the level of anonymity with the anonymity set sizes: how many user records share the same properties, which could make them identifiable. (And here is the point where differential privacy fans raise their voices, but that story is worth another post.) However, this is much harder if we think about highly dimensional datasets where you have hundreds of attributes for a single user (think of movie ratings, for example).