The Elvish this generator uses is called "Common Elvish". It is a fan-language developed as a supplement the TTRPG "Dungeons and Dragons"! It's made to be used in your world, for character and place names. Even titles.
But that's the meta version of the story. What about in-world? A little lore never hurt anyone!
Common Elvish is a language spoken by almost all elves. It's orginal comes from the dawn of Elves themselves, though it has developed and changed since. It works as a lingua franca between the different Elven cultures. It is a language for trade, research, and history.
But no elf would consider it to be their mother tongue. Every clan or culture of elves speaks a dialect of this language. Some are similar enough to be understood, without using Common Elvish. While others are enough changed that they are, at this point, seperate languages. A good example of this is Drow Elvish, itself full of various dialects.
That being said, Elves will usually choose to name themselves in Common Elvish. There are exceptions, of course.
Common Elvish is maintained through the generations, mostly unchanging, due to Elven reincarnation. Elves, in their trance (also known as reverie), often relive past memories. As children, these are memories of their past lives. Knowledge of Common Elvish often comes through the reverie, and it's expected a young elf will simply come to understand the language without being taught.
Of course, for the half-elves of the community, this represents a problem. They might find it frustrating how little information they are given. Since Elvish really have no culture of teaching Common Elvish. Most books written in Common Elvish assume an adult's reading comprehension. Thus, they might from that they must learn the language from bards or scholars outside their clan, who are usually members of other races - or half-elves themselves. So while their community might harbour no ill-will towards them as half-elves, they may still experience challenges and prejudice.
When it comes to histories, songs and other records, Elves will use Common Elvish only for the things they deam revelant to others. Scholarship, for example, uses Common Elvish exclusively. The same goes for the magical or martial arts.
However, when it comes to stories and knowledge relevant only to themselves, it is usally recorded in their own dialect. And often, no recorded at all. Instead, elves usually favour oral history. With their long lives, it makes far less sense to pen down information for prosperity. An elder elf, for example, is by their nature a library of information about their homeland. From the flora and fauna, to the weather patterns, to the individual history of the people who live there. This information they can directly impart, often through song, to new generation for hundreds of years.
Why is this important? Well, outside of flavour - feel free to alter names the generator creates. It can just be a dialectal variation!