Sidebar blocks should use a common template suggestion, similar to regions.

In general, I'm not in favor of region-specific (or context-specific) styling. A single class should always present the same styling no matter where on the page that class appears.

If a specific block needs different HTML, you should use a theme hook suggestion specific to the block, not specific to the region.

If you want to add a class to all blocks in a region and that class provides a consistent styling, then the theme hook suggestion you proposed would be useful. But that is an edge case that you should implement for your specific sub-theme. :-)