That’d be my guess. I have a friend whose whole immediate family call him Mikey. Now, he should have spoken up sooner (he’s “all growed up”, but only started asking them to call him by his real name in the last few years).
Everyone else in the world calls him Mark…
(Michael is his middle name!)
I presume he wouldn’t be fond of “Mikey Mike”.
My mother wasn’t perfect, nor was our family life, but I can’t recall my mother ever talking like that.
Whenever it was, it was young enough that I don’t remember. I did with my kids until they were about 2 or 3. But as I remember it, it was less a matter of me speaking in the third person and more a matter of avoiding pronouns. IOW, I didn’t just refer to myself as “mommy” instead of “I” or “me” , I also referred to the kid by name, rather than “you” and to other people by name instead of “he” or “she”. Of course , I didn’t avoid pronouns completely but I used them was less than I ordinarily do.