Because it’s how one says ages in french.

We can revert the question:

Why is to be used instead of to have when expressing/referring to age, in english?

Just to get a feeling for why such a meaning is possible, you can take it as meaning

I have 16 years (of age under my belt).

or think of it as similar to the past perfect tense in English (have+past participle), which links past and present

I have (aged for) 16 years (and this is me now)

If we take “copula” as the fancy term for “to be”, and look around other languages:

• French “to have”: “J’ai N ans.”
• Spanish “to have”: “Tengo N años.”
• Italian “to have”: “Ho N anni.”
• Latin “copula”: “N annos natus sum.”
• English “copula”: “I’m N (years old).”
• German “copula”: “Ich bin N Jahre alt.”
• Japanese “copula”: “十六才です。”
• Korean “copula”: ” 열여섯 살입니다”
• Chinese “counting word only”: “我16岁” (literally, “I 16 counter-for-years-of-age”)
• Russian “dative”: “Тебе 16 лет.” (literally, “to you 16 years”)

So it looks like Latin-influenced language prefer “have” while saxonic languages prefer the copula, but globally the expression can be anything.

Last thing, you can use “to be” in French:

Elle est âgée de 53 ans.

but it sounds journalistic.

(NB: I had to look up some age expressions)

Peut-être l’expression était “J’ai l’âge de seize ans”.