Does variance 'change' when we compare 2 groups?

Yes [and no], depending on the model we believe in. See e.g. below:

1.A 2 sample t-test compares the difference between means against zero

2.ANOVA uses an F test to compare within-groups sums-of-squares (pretty much how far away individual scores are from each group’s mean) with between-group sums-of-squares (how far away group means are from the overall/grand mean)

3.SEM can compare the means using a chi-square test for the 2-group model

ØALL these 3 methods allow for variances different or assumed/set equal; the last method only estimates [from the 'equal variances model', which may fit better/worse than the 'variances different' model] a new (common, equal)  population variance in addition to the 2 population means, i.e. the model yields 1 'NEW' variance, that may differ little/a lot from the 2 sample ones. These differences are taken into account and ‘reported’ along with differences between sample means and estimated means, as part of the FIT of the model. Good fit=ok, bad fit=not ok.

What is a 'variance' after all?

Is it 'set in stone' once we gather data from a sample? Yes and NO, yes in one instance primarily: when we look only at 1 variable , in 1 group only.
If we have 2 groups OR more than 1 variable, say males & females, we can compare the 2 variances, and that means in fact comparing a MODEL with the 2 variances different to another model with them SET equal. The 2nd model here will yield NEW [population] variance estimates for males and females, not surprisingly equal (WE set them =!).