My only anecdata is that I have only noticed it in the past few years, and it makes my skin crawl. Don't know how to get good data on the extent or prevalence of this practice, sounds like a lot of work.
I'm describing an extreme, but I reckon it's the case that in American society there has been a significant increase in the use of animals for emotional work. As in treating domestic animals as pets and members of the family rather than work animals (for deterring vermin, hunting, guide work, &c.). Referring to pets by familial terms is a linguistic reflection of that.
I encountered it first in the 1970s -- when my age was in the single digits. It might have been new then, but few people seemed to think it was noteworthy, so I doubt it. So, I think "have begun" is somewhat of a misrepresentation.
> Don't take an animal and twist it into filling your own emotional needs by treating it like a human infant.
IME, virtually none of the people who use this language treat their pets "like a human infant", and many of them have either biological or adopted children as well as pets.
If you want kids, have kids or adopt them. Don't take an animal and twist it into filling your own emotional needs by treating it like a human infant.