This seems like the most basic question in the world, but damned if I can find an answer.
Is there a keyboard shortcut, either native to Visual Studio or through Cod
Nothing I'm aware of, but writing a macro to wrap it in whatever tag you want shouldn't be hard. I have a similar one that will wrap my selection in a region block.
For those who use Visual Studio 2017: Right click on an html/cshtml area, or select some elements to wrap, there is a Wrap With <div> button on the list.