Does it seem odd to you or is it just me?

In the UK it is a construction used by working class men in the North or England particularly. The wife, the missus..etc. It's dialect more than anything more I think.

Maybe they do not want to make it seem like they're a possession. 'My' has a possessive connotation.

Ooooooooh. Thats a good one. Its a meams to distance ones self from the person. Like when Bill Clinton was saying, "I did not have sex with THAT WOMAN". Of course, it could just be slang. But, I tend to think everything has a deeper meaning. I bet it's because they are feeling distant.

Cops also look at this when interrogating people. Like, if a husband or wife murders their SO, and in the interrogation, they say, "that person," it shows a lack of affection or caring that they are dead, which gives some degree of motive.

Me too. Me and my siblings always get referenced as 'the kids' around my fathers friends. My mother as well.

"Yeah, because I gotta feed the kids."

"Yeah, the wife wants me to do that."

In Dutch men usually say "die van ons" which would be in English "Mine" as we're somekind of property/possession. How would it sound like when a woman would say "The man has bought a gift for me" LOOOOL it sounds hilarious. But if it would be a gift, we (women) would probably more say "My sweetheart" or "my hubby" I guess ;)