Is there any specific "Sampling Design" that considers the geographical characteristics of the region/area in the study of social sciences, if the region is considerably driven by its geographical setup?
If you have auxiliary/regressor data, that could help, but I'll assume here that you only have enough other information, if any, to separate data into categories or strata, and use strictly randomized-probability-of-selection, design-based sampling and estimation in each:
If you want to study each geographic region separately, you can sample them each separately. This is not really stratified random sampling because in that design we are trying to improve the overall estimate, or obtain the same overall accuracy with a smaller sample size, and do not care about publishing each subpopulation/stratum/category - just the more aggregate level.
If you want to get an overall result while making it easier from a logistical perspective, cluster sampling can help.
Area sampling means your frame could be geographical areas rather than a list frame. As in cluster sampling, you draw areas and then see what second stage units are available in them.
My guess is that you want separate random sample designs within each category of geographic regions, i.e., subpopulations, because you want to compare them - but I may be misunderstanding your question.
When finding different categories/subpopulations/geographical regions to study separately, you will need to decide what geographical characteristics will be used to define each such region, based upon what you want to study/compare.
Yes, there are sampling designs which use of the administrative registers. They are classified by states, regions, cities, etc. Each country has its laws to the access to these registers or censuses. These would be the first pass to design a frame of the population, and then to select a sample of it.
Sampling design is how exactly you are going to perform your research. Your method, in other words. It can be of course very specific depending on what you are doing. For illustration I attach my example: sampling design of the seagrass sampling in the Mediterranean Sea: it is how exactly I was doing observation of the seagrass (the size of the sample matte - one meter, then a marker, then another seagrass matte, how long, how broad, etc). My topic was very specific, but may be it can serve you in terms of ideas and how to draw it in general.
In social studies, sampling could be for example, "studying education level of the population living in the eastern part of the country and compare it with group of people living on the west (how many of them attended university there and here)", just for example. Well, something like this..