Yes, this is used in the United States (and elsewhere in the English-speaking world). It's hard to say exactly how popular it is, but you can certainly find plenty of examples on the internet.
Note that "Baby LastName" is also used when the gender is known. "Baby Girl LastName" and "Baby Boy LastName" are also used like this, notably in some hospitals:
There is a very simple reason why babies go unnamed. Most of babies are born fast and parents can take weeks or months to pick that first name. Simply put, the delivery room and the receiving nursery can’t wait for a parental decision when the new arrival pops out squalling. The baby immediately needs weighing, Vitamin K shots and a medical exam to assess whether the baby is healthy or sick. Immediately, the clerical staff produces labels declaring that Baby Boy Smith or Baby Girl Garcia has arrived and they label the medical record accordingly.
Babies Without Real First Names At Risk
The form "Baby LastName" isn't new either; I found an example from 1892:
So is ushered into the world Baby Jones, very red and hungry, and very much troubled with insomnia.
Ballou's Monthly Magazine
Here are some more examples:
A friend of DD's, who had her first last week, called the unborn child Baby Smith, but with their name of course. —Breezy_Carol
Code names or nick names for unborn babies
I believe I agree Mahn she has more than enough money to get her own stuff they should donate money to charity ob behalf of baby Simpson. —Stephanie
It’s their surname, so lots of people refer to their unborn children as ‘Baby Johnson’ or ‘Baby Smith’ or ‘Baby Jones.’ —Melody
Jessica Simpson: ‘I’m So Excited to Have a Son’
To others they are baby smith (SOs last nane) or now my two littlest angels. —CassiJai
How to grieve when baby's sex is unknown?
In 1989 I spent the summer in Burlington, Vermont. One day, while driving around town, I came upon an old cemetery. It had a section reserved for babies, and all of them had died before the turn of the century. Often the children had no first names. The tombstones read simply Baby Johnson or Baby Smith.