My wife is absolutely fine in my books. The same way you can say my friend or my acquaintance or what have you, you can say my spouse/wife/husband/whatever without implying any sort of ownership. My view might be skewed being an ESL, but the same applies to German, Russian, Ukrainian, French, for example, which I speak too. I've never heard people complaining over those usages
"My" can, but certainly doesn't always imply ownership. It implies that the qualifier for person B in the sentence is applied to their relationship to me, person A.
My banker, my hairdresser, my dentist, my accountant, my contractor, my neighbor, my boss, my elected official, etc.
Probably not at all on you, here, but this is a good example of "exaggerated progressivism" or fake-woke/politically correct speech. It weakens the credibility of the progressive movement and gives conservatives silly exaggerations to point to and mock, when this kind of stuff emerges.
People are stupid and looking for reasons to be offended. Usually it means their opinions are to be ignored as they simply exist to argue about non issues.
When I read "the wife", I read it as this. It feels impersonal, like you are intentionally distancing yourself from your spouse. Like, why would you do that unless you are literally Al Bundy?
I don't know if "my" always means owning the noun it could mean co-owning the relationship. If I say "my partner" or "my kids" or "my job" I don't think most people think I am a slave owner who somehow has an unbreakable contract with work.