I have 8 IV and 1 DV, they are all dichotomous and categorical data, so I am using Cross-tabulation Chi-square test. So 8 tests conducted. All test have at least one expected count less than 5. So I am interpreting fisher's exact test and not Chi-square. 3 out of 8 has fisher's exact test p=1. The groups are not identical. And the p value of chi-square of those 3 test are different from each other (although they have the same p value for fisher's exact test which is 1). I think there is something weird going on. Why 3 tests have p=1? What could be the problem?
A p value of 1 means you cannot reject the null hypothesis of no association. The value 1 is not very special for an exact test and a small sample.
Note: When doing multiple tests, you might want to consider correcting for type 1 error inflation, depending on the setting.