EE cores are named with their dimensions.
The following is from EE10's datasheet from TDK:
As you can see, "10" here indicates the width in millimetres of a half core.
Correct Full name includes other dimensions as well. For example, when you see an EE10 it can be EE10/5/5 or EE10/5.5/5 (depending on the manufacturer) which includes the height of a half core (5.5 mm) and depth (~5 mm).
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 such as the ones I mentioned above.