I understood that the variable CPSIDP is an unique identifier for each individual. I have seen in the data that there are individuals with the same CPSIDP but different birth place for example (they match in CPSIDP, age, sex and race). In which cases is this posible?
Due to the complexity of linking persons and households across CPS samples (e.g. migration, mortality, non-response, and recording errors), erroneous links can sometimes occur. Additionally, even accurate linkages can result in inconsistent responses across samples, due to errors on the part of the respondent or in recording the response. The inconsistent birthplace responses you have identified are likely due to a combination of both erroneous links and respondent/recording errors. Ultimately, it is up to the individual researcher to determine the acceptability of the linkages made using CPSIDP. I recommend this paper on the process of creating CPSIDP and the inherent complexities of linking respondents across months.
Hope this helps.