The correct answer is Barnum Effect
To begin with, the Barnum Effect has several names. From Forer Effect and Subjective Validation to Barnum Effect, Forer Effect and Subjective Validation (the latter being the original terms in English, commonly used here as well).
By definition, the Barnum Effect chronicles the mental tendency that we all have, as human beings, to take generalist personality descriptions as having been made specifically for us.
Barnum Effect is a type of cognitive bias. This means that it is part of a huge set of errors made by the brain, which, when processing the information received from the world around us, takes wrong conclusions as correct. They seem to make a lot of sense at first, and to some extent, that is enough to understand our context and survive.
It is not necessary to contain the term "bias" in the name to be characterized as one. Although the faded emotion bias, the omission bias and the social desirability bias, already deliver gold right away, many others (such as heuristics, illusions and effects) are part of the same group of psychological phenomena.