I don't think this has changed since Word 2010, though I recently updated to Word 2016. When you bring up the context menu, does it have an item in it for "Edit Alt Text"? If it doesn't, what are the options?
My one machine that still has Word 2010 on it is currently being cranky about restarting.
Brian - Windows 10 Home, 64-Bit, Version 1809, Build 17763
Explanations exist; they have existed for all time; there is always a well-known solution to every human problem — neat, plausible, and wrong.
~ H.L. Mencken, AKA The Sage of Baltimore