Is there a relation between chi-squared test and logistic regression model in a similar manner between ANOVA and linear regression?
If the setting is a $r\times 2$ contingency table, crossclassifying a binary variable with an (unordered) factor with $r$ levels, you can use binary logistic regression or a chi-squared test, and you will get very similar answers.
The logistic regression will give the likelihood-ratio test, and the Pearson Chi-square is the corresponding score test. Details: G-test vs Pearson's chi-squared test. For an $r\times c$ contingency table (both factors unordered) much the same will be true replacing multinomial logistic regression. So logistic regression can be a good starting point if you need some extensions/generalizations to these chi-squared tests.