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Everyone getting a power is Everyone Is a Super
Also One Person, One Power when its specifically one special trait.
Good examples are Xanth, where everyone is born with a single magical talent (running the full Superpower Lottery gamut from from turning small rocks green to the ability to Baleful Polymorph anyone in arm's reach at will) and My Hero Academia where a significant percentage of the population is born with a Quirk that gives them a single superpower, frequently of the Heart Is an Awesome Power variety.
A work of Speculative Fiction takes place In a World… where every person has a distinctive thing - a different superpower, a Mon, a weapon, a tattoo, whatever. The main character is distinguised because his thing is extra-special (the only golden weapon), or because he's the only one who doesn't have the thing.
Most times it's all about magic, but it can also be a cultural aspect.