This is an interesting question. I think probably to get a reliable answer you might start by posting an example of a data table, including what the rows and columns represent, how you calculated the odds ratio, and how you calculated the confidence interval.

My suspicion is that odds ratio and risk ratio don't "match" the chi-square test of association in the way you are considering them. Usually phi or Cramer's v is the appropriate effect size statistic for a chi square test of association. It may be that if the effect you want to describe is captured by odds ratio or risk ratio, that another statistical test may be more appropriate.

But there may be other reasons for your somewhat conflicting results. So posting a concrete example is probably best.

Yes, you should post the data.

From reading your post I would think that the CIs are calculated as inverted tests. But then what you observed should not happen. Maybe the software uses different algorithms that could lead to numerically [slighly] differeing results, or the CI is not calculated as inverted test. Checking your data with different software might help.

Is the chi-square test that of contingency table? Probably. I'll base my answer on that assumption.

The reason for the discrepancy is that although Odds ratio is an effect size measure of the difference of proportions, the test of significance for OR tests the null hypothesis log(OR)=0 and uses a normal distribution.

The chi square test of proportions uses the chi-square distribution.

Although most of the time when the proportions difference is large, both will show a similar answer. But in marginal cases, there may be a discrepancy because the two tests use different statistics and distributions.

I would suggest not using both the approaches together. Better go for binary logistic regression models for each pair of variables and report the coefficients and ORs based on that.