blessings:
Games where the protagonist gets a birthsign blessing

TES III: Morrowind (Nerevarine)
TES IV: Oblivion (Hero of Kvatch)

Games where the protagonist does not get a birthsign blessing

TES I: Arena (Eternal Champion)
TES II: Daggerfall (The Agent)
TES: Battlespire (The Apprentice)
TES: Redguard (Cyrus the Redguard)
TES: Stormhold (Master Tunnel Rat)
TES: Dawnstar (Hero of Dawnstar)
TES: Shadowkey (Soul of Conflict)
TES V: Skyrim (Last Dragonborn)
TES Online (The Soulless One)

As you can see, receiving a birthsign blessing is something of a rarity, even among the movers and shakers of the Elder Scrolls universe. Before we examine those who got them, let's review what the constellations actually are.
The Celestials
In The Elder Scrolls Online, it is revealed that the constellations are actually powerful god-like beings named the Celestials. During the games, the constellations actually fall from the sky and takea mortal form, at which point the Celestial Serpent plots revenge against them. Eventually, they do all return to the sky.
I imagine that, like the actual gods of Tamriel, they have the power to directly influence the world, but choose to do so to a limited number of people. So let's look at the two they did influence.
The Nerevarine
The Nerevarine (the hero of Morrowind) is the reincarnation of the Chimeri warlord Nerevar and is tied to astronomy in multiple ways. The prophecy "The Seven Visions" says the following of him/her:

On a certain day to uncertain parents
Incarnate moon and star reborn.

Nerevar also went by the title "Nerevar Moon-and-Star" (which is what "Incarnate moon and star" refers to) and it's clear that the Nerevarine will be known to be born