Why the structure "was born", and not "is born" like in many other languages?

The apparent similarity of "Je suis né" and "I was born" is deceptive. Even though both use a "to be" auxiliary, they are actually not very similar structures in terms of how they fit in the rest of the language's grammatical system.

English "be born" is a verb that takes the form of a passive construction, despite not actually being a passive construction ("to be born" can't be converted to an active-voice equivalent, and it can't take a by--phrase to mark the agent of the action). Historically, born is derived from the past participle borne of the verb "to bear", used as a passive particle. Compare a true passive like "to be killed": "I am killed" refers to the present, and to refer to the past, the auxiliary is inflected for past tense: "I was killed". (Or to refer to the past in other ways, you can use a perfect construction: "I have been killed", or "I had been killed" for the pluperfect.)

French "Je suis né" is a perfect construction. In French, as in English, the perfect construction consists of an auxiliary verb and a past participle; the auxiliary is not inflected for past tense unless you want to refer to a "past-in-the-past" (pluperfect) situation. So "J'étais né", with past inflection marked on the auxiliary étais, means "I had been born".

French mostly uses the auxiliary avoir "have" to form the perfect, but a handful of verbs such as naître "to be born", as well as all reflexive verbs, use être "to be" instead when forming the perfect. This use of être to form the perfect is not very comparable to the English use of "be" in the English passive construction, or in the pseudo-passive construction "to be born". The French usage of être to form the perfect is comparable instead to the obsolete usage of "be" in English to form the perfect of verbs like become (as in "He is become my salvation").

Unlike in English, the French perfect construction is used to express a plain "perfective" past, with no special connotation about a relationship between the past event and the situation in the present. Since "Je suis né" is a French perfect construction, the word "suis" doesn't actually mean "is" here, even though that's the translation of the word suis in isolation. So I disagree with the comment that suggested thinking of "Je suis né le 13 Novembre" as meaning something like “I have November 13 as my birth date”: it doesn't mean that, it means "I was born November 13". You can use sentences like "Il est né en 1943" to refer to someone who was born in 1943 and has died since: the "suis/es/est... né(e)" construction does not have to describe a currently existing person.

Naitre is used in the present tense, where it inflects the same as other French verbs, rather than looking anything like a passive construction: e.g. je nais "I am born", vous naissez "you are born". English happens to lack a verb that can be used as a single word in the active voice with this meaning.