It is the same word.
If you mean 'proof of identity' it is 'une piece d'identite'.
Personal Identification Number
That would be Projective Identification, or just Identification. It is one of the classic defense mechanisms.
Yes, a "gendarme" is a police officer in France who is part of a military branch responsible for policing in rural areas and small towns. They have both civilian and military duties.
pathogens
"Le mal de mer" is the French term for seasickness.
The French term for the enlightened thinkers was "Les Philosophes."
The French term for 'umbrella stand' is 'un porte-parapluie' (plural: des porte-parapluies) in French.
There is no French term for spatchcock in French. It is not a usual way to cook or grill chicken in France (it would be dismembered or broached).
Primary identification or socialization
she is French means she has French citizenship, she is a citizen of France.
The official term for French speaking countries is francophone countries.
abandonner is 'to give up' in French.