The title is usually given by a registry to a stallion from that registry, and that is approved for breeding by that registry, and that has met whatever additional criteria the registry sets (performance success and/or proven success as a sire).
In the case of the German registries, it is not a title given lightly.
It seems that all AHS stallions are called Elite (apparently a convention of the AHS). My understanding is than a German Hanoverian Stallion earns the title Elite with an SPT score in excess of 120 points.
As said before, it depends on the registry/Verband.
Trajehners for example get the Elite title in Germany based on their own success as competition horses (has to be advanced level in any discipline), and/or in combination with the success of their offspring (performance tests, competition for older get). Much of it is also based on the FN breeding index systems. Before a stallion becomes an elite stallion in this breed, he is placed on a watch list. Some make it off that list and get the full title, others don't.
However, at the end of the day it is just a title and means little to nothing. most breeding index numbers are flawed (very shaky statistics) and the requirements for elite status are not always as transparent as one would like. For example, the fact that some stallins have 1000 foals while others only have 50 is not factored it - obviously, the higher the amount of offspring the better your chances of hitting the nail occasionally. Personally, I don't pay any attention to titles if they're not based on sport results.
What sparked my interest was an article I saw on eurodressage re. Trakehner stallion approval where some of the stallions (e.g. Gribaldi, Hohenstein (sp?), Biotop) had "E.H." prefixes and some did not. I do not recall seeing that prefix before in other similar articles.