Impa Is Not A Name and Related Timeline Shenanigans.
While it can be argued as coincidence that Zelda and Link’s incarnations all share the same name (with the exception of Hylia and Tetra), with what we know of Impa and the Sheikahn roles in protecting Zelda throughout time, I can not believe her ‘name’ is by chance.
Impa is the one Sheikah chosen out of her tribe to act as Zelda’s personal protector and nursemaid in most of the games she appears in. She (in all her incarnations) is one of the only two canonly recognized member of the tribe we ever meet (the other is Impaz, not Sheik.*) while the rest remain hidden. While Kakariko is stated to be the Sheikahn village, we see no one else dwelling within that looks like they could be a Sheikah during the events of Ocarina of Time.
The Sheikah are known to have many prophesies that they pass down generation to generation. They know of certain events that should happen at certain times - but they don’t know all. So an Impa is likely sent to serve Zelda directly when a time of known prophesy is at hand. This would explain the absences of an Impa in some of the games (while the total death of the tribe would explain it in others, depending on the timeline post-OoT).
Even if Impa is named ‘Impa’ at her birth, she is likely given this name because she has been chosen by signs present at her birth to be the one trained for the role and then sent to aid Zelda. In this regard, given what we know of the Sheikah and their dedication to their duty, it can easily be extrapolated that Impa is more a title than actual name. The one who bears it giving up their individuality in history to the greater service of the Royal Family and Zelda in particular.
Of the three timelines, only one ends up preserving this role: the Downfall Timeline. Impa appears in all of the Downfall Timeline games except A Link to the Past and Link’s Awakening. She is the nursemaid and protector of Zelda, as well as a source of ancient tales and prophecies to aid Link in his quests.
In the Adult Timeline, while we see signs of the Sheikah’s former presence in Wind Waker in the Forsaken Fortress (it has the Sheikah Eye decorating it at points), and the piece of Gossip Stone that lets the King of Red Lions speak to Link bears the Eye as well, there is no Impa. This could very well be because the Sheikah - who were already close to dying out in Ocarina of Time - did not survive the Flood. It could also be that even though they did survive the flood, they could not trace Zelda’s bloodline to the Pirate Queen Tetra. However there was no evidence of their continued existence in Spirit Tracks, so until another game takes place in that timeline, best guess is that they died out.
In the Child Timeline, there is an old woman who is one of the last of the Sheikah. She bears the name Impaz and said she was named in homage to the town’s founder: Impa (a different Impa who lived before the events of Ocarina of Time). Her role and duty as the last remaining resident is to wait for the “messenger to the heavens” and give him the Ancient Sky Book. Her duty was not to serve Zelda directly.
There is also one other possible Sheikah or part-Sheikah in Twilight Princess: Madam Fanadi, the fortune teller. She shares only the distinctive red eyes of the tribe, but has apparently tattooed the Sheikah Eye to her forehead (or has a very good makeup artist) - likely because it is a symbol known for helping one see through illusions. There are good odds she is half-blood Sheikah (at best) due to the tribe being almost extinct: we know all but Impaz left the Hidden Village (old Kakariko) and breeding outside the main lines with Hylians would at least keep the blood alive.
*Due to Sheik being a cover for Zelda, s/he does not count as a Sheikah for the purposes of this discussion.