Given all the evidence (and lack thereof) from the books, I'd suggest it's an honorary title. If it does have an effect on the person in question, it would be at a subconcious level. Maybe they walked a bit taller, a bit more self-confident, knowing that an Elf has considered them special enough to name them Elf-friend. Goldberry is a special case, with powers unknown, so what she can and can't detect is tricky to determine an answer from.
We do see a number of characters named Elf-friend, as if it had specific significance, Frodo as you noted, but also Aragorn in the Fellowship of the Ring:
'Eight,' said Legolas. 'Myself, four hobbits; and two men, one of whom, Aragorn, is an Elf-friend of the folk of Westernesse.'
And Gimli in the Appendix A:
[Gimli] was named Elf-friend because of the great love that grew between him and Legolas, son of King Thranduil, and because of his reverence for the Lady Galadriel.
Anardil, Meneldur's son, was also mentioned as the greatest of Elf-friends in The Mariner's Wife in Unfinished Tales:
Ereinion Gil-galad son of Fingon to Tar-Meneldur of the line of Eärendil, greeting: the Valar keep you and may no shadow fall upon the Isle of Kings. Long I have owed you thanks, for you have so many times sent to me your son Anardil Aldarion: the greatest Elf-friend that now is among Men, as I deem.
It's also used numerous times in the Silmarillion, first to indicate the original Men and then later the "true" Numenoreans who did not fall under the thrall of Sauron, as detailed in the glossary:
Elf-friends The Men of the Three Houses of Bëor, Haleth, and Hador, the Edain. In the Akallabêth and in Of the Rings of Power used of those Númenóreans who were not estranged from the Eldar; see Elendili.
Elf-friend was presumably not just a once-off bestowal by Gildor then. Its original meaning was literally "one of a race or class who is a friend of the Elves", i.e. the first Men or Elendil's Numenorians. Over time it appears to have become more restricted and logically so, as Men and Elves no longer had the close contacts that they once maintained. It may have still applied to the Dunedain of the North, given they still maintained a relationship with the Elves, and Aragorn's naming as such may have been because he was a Dunedain rather than describing him specifically but that is my own speculation.
That aside, it also has obviously been extended depending on the circumstances of an individual, i.e. Frodo and Gimli. None of the other works go into more detail on how it applies at the individual level, but given it was originally applied to whole classes of Men it seems unlikely that some charm or such is directly applied - especially in Tolkien where magic is not a day-to-day act.