Is collinearity and lack of complete overlap the same when dealing with categorical input variables?

Hi all,

I have a conceptual statistics question. I was reading about lack of complete overlap between treatment and control groups, defined as follows: " Lack of complete overlap occurs if there are regions in pre-treatment variables where there are treated observations but no controls, or controls but no treated observations (Gelman and Hill 2006)." according to a quick google search.

I’m wondering about the case of multiple categorical input variables. Let’s assume some of the values of variable x1 only occur together with certain values in variable x2. I guess this would be collinearity between these two variables? However, I’m wondering if it isn’t also lack of complete overlap, because not all combinations of values of x1 and x2 are present in the dataset.

Can anyone confirm/refute this and/or provide some better intuition here?