EE cores are named with their dimensions.
The following is from EE10's datasheet from TDK:
As you can see, 10 is the width of a half core.
Correct naming includes other dimensions as well. For example, EE10's correct name is EE10/5/5 or EE10/5.5/5 (depending on the manufacturer) which includes the height of a half (5.5) and depth (~5).
So yes, EE10, EE15 and similar names indicate the core shape and dimensions. But always check the manufacturer's datasheet as some of them may have slightly different dimensions (just like the example above).