Maybe the 'soul name' is just a device that the Wolfriders evolved in order to protect their 'innermost selves'... maybe the elf part felt it needed protection from the wolf part? And maybe it's unique to Wolfriders. Maybe it's a key to their minds/inner selves that is specific to them?
And maybe elf names, for other elves, are just names...
That is what I am thinking too. In the ancient world, for instance among Jews ab 1000 BC, names usually described a persons inner quality. (Abram went to Abraham to describe his contact with Jehovah, "h" taken from "Jehovah", etc)
The elves are ancient. They behave like ancients and they are thinking like ancients and all their names describe either their personalities or their physical traits, just as the ancients. Their soul names are simply descriptions of their inner spiritual life, written in their own elvish language. They reveal their harmless aspects through their English names, and they reveal what they have to protect through their elf names.
Cuttter did not find a soul name in Leetah, since she doesn't have an animal nature like a wolf. The soul names seem to be connected to their wolf nature, and they probably describe the elves own animal essence or peculiarities. Other elves may find this essence disturbing since they may not be able to relate to it. The Wolfriders are also dependent of living as a group, and if everyone suddenly knew their peculiar animal own, and the full meaning of it, there could be conflicts and the group could then fall apart.
The list of all the soul names in EQ are probably a catalogue of terms describing the elves animal aspects, something others, or the non-animal, my find frightening, in other words it is dangerous for the elves to reveal it, since the non-animal from the "High Ones", or humans, may hate it. (The Gliders also hated the Wolfriders. Most humans hate the Wolfriders etc.)
They are only able to recognize each others soul name through the recognition of it. If I remember correctly Skywise and Cutter recognized each other when they where fighting and both where showing an equal amount of animal life. Their inner animal own could then be revealed without any conflict since they both took part in the same kind of life, (whatever that could be) on an equal level, and they saw each others animal own, and the moment they saw it they also named it, and those names were their soul names. (I really wonder what they mean). Since they were on an equal level they could also understand each others own, and they had no reason to hate it.
Leetah was first repelled by Cutter until she saw aspects of Cutters personality and named it "Tam". She could then fell in love with him, since she could not find anything else, or something worse. "Tam" could for instance mean some kind of protection instinct derided from the wolf, and Leetah found it attractive. It could also be something else. Cutter searched for a soul name in Leetah but did not find any, and naturally he did not worry about it. She only had a name, describing something not related to the wolf nature, but Leetah could also be a soul name.
We know what a "Sun-Toucher" mean, but not "Leetah". In the ancient world Sun is very often the symbol of love. We also know what Cutter mean. Leetah doesn't mean anything in English and it could then be seen as a soul name itself, since the meaning is hidden.
The point is: When the elves perform a certain act they may reveal their soul names, and if the other elf is able to recognize it, the hidden secret essence may become harmless through the knowledge of it. If you can name something, you can also see how it differ from everything else, how it relates to everything else, you can understand it, perhaps control it, and all this can drive anxiety away. Perhaps both elves partake in the same kind of inner life. Two elves can then get into close contact with each other without being afraid of their own being, or animal being. In recognition they simply know that their secrets is not anything to be afraid of.
The alternative is that they do not recognize it, and they will then not be able to see what they are not able to control, or relate to in a good way.
All the elves' soul names are also revealed during, or right after, some kind of action scenes, sometimes extreme ones.
A soul name is after all connected to the soul, and soul names will be meaningless if they didn't do any attempt to describe the soul. You would then only have a word without content, and that is not something elves would find important. Very often they prefer to send, instead of talking, and that also tell something about their fondness for mere words.