Custom site variables are specific to each site.
This is why you may encounter issues getting data from specific custom site variables when switching from site to site in Dashboards or Data Query. This is the case even when a similarly-named custom site variable with the same data type is present on the other sites you're looking at. Our system considers them to be wholly distinct.
When you switch to another site in the interface, you thus need to reselect the variables for said site.