Nah there isn't an "identity function" for words like that, nor can a single language "claim" a word. When a language fully assimilates a word with its origin in another language, it also becomes fully a word in the assimilating language as well. The connotation, denotation, pronunciation, and/or orthography can each change, or not, in either language or both, and it is a different word from that point on regardless.
Your point about "genetically french" is also wrong in almost exactly the same way so it provides an illustrative comparison. There's no single french ethnicity, just people who are french and some observable trends. Someone is french who has french citizenship, a word is english when it is widely spoken by native english speakers.
I also need to point out that this is not a matter of opinion. Linguistics is an established field and consensus is clear on this. You are confidently asserting things that are well outside the mainstream of the science.