Dh has a younger sister with the same first name as me. Mil always refers to her as “my” (name). Emphasis on the my. Yes shvigger dear I know you love your daughter. No need to rub it in.
I have a sister and sister in law with the same name. Also a brother and brother in law. As well as various nieces and nephews. Somehow we all manage without "mine" or "yours" lol. It really isn't that confusing.
The use of "my" comes across as a bit, unaware, maybe? Kind of the way some people refer to their parents as "Mom" when talking to others not in the family. Just stands out as a bit quirky to me...
To judge someone as emotionally unhealthy based on a single speech pattern (and apparently only because you personally find it annoying) is as fair and correct as deciding that a person is drunk because you observe him throwing up into a wastebasket.