In Japanese we literally call them “egg stands” and I always believed that that was the English name given for this object but after some googling I found out that an “egg stand” seems to imply something that holds an ornamental egg for display purposes and there is “egg holder” which seems to be used more for the thing that eggs are sold in in supermarkets and the thing you serve eggs in is called an “egg cup”.
Anyway, here is something new I got last week because I am currently on a boiled “egg phase” for my breakfast.

The “L” side for a larger egg. Most eggs in Japan are labeled “M” or “L” for their sizes. Yes, the egg “stand” is Japanese made. Very slick design.
It’s a “stand”. There is nothing “cup” to it. 😀