The test you get with chisq.test is for counts - used to compare proportions or test for independence with categorical data, that kind of thing.
On the other hand, t-tests are usually for comparing means.
There is a test involving variances (a one sample variance test) with normal data that is a chisquare test but you don't get that test with that command.
With two samples and normal data there's a corresponding ratio-of-variances F test for testing equality of variances, but it's generally not recommended (it's not robust to violations of normality). Levene or Browne-Forsythe -- or a few others -- are more often used, typically corresponding to a form of ANOVA on deviations from some measure of location.
When those deviations are bigger on average it would correspond (under some reasonable assumptions) to the variances being bigger.
An equivalent to Levene or Browne-Forsythe could be performed with two-samples (on deviations from the mean or median, respectively) and could even be done as a t-test rather than an ANOVA.