This is a misunderstanding.
Ich bin geboren.
is not Perfekt Aktiv (perfect tense, active voice), but Präsens Passiv (present tense, passive voice). The Präsens is used talking about past events, but that is a use case for Präsens in German.
In German, this is the idiomatic way to express this.
Ich war geboren.
is not idiomatic, and German differs from English here.
German has two different passive constructions: the passive voice with sein, called Zustandspassiv ("passive voice of the state") and Vorgangspassiv, the passive with werden ("passive voice of the process"). Zustandspassiv is mostly used to focus on the result of a process, while Vorgangspassiv shifts the focus on the process itself.
geboren is the participle of the verb gebären, "to give birth".
You can see that it is used in passive voice here, because gebären in active voice is using the auxiliary haben. So the active perfect tense counterpart of the sentence would be
Ich habe geboren.
and it would mean "I have given birth".
The passive construction in German means as much as "I was given birth to", but that is not proper English of course.
I think the English I was born is also a passive construction, but I am not so sure about this.
The correct past tense of the passive voice would be
Ich wurde geboren.
In my feeling, this constructoin is idiomatic in German if you are talking about other persons, especially about historical persons.