Not even close.
The Teddy Bears had their one and only hit in November of 1958. That means they were after the heyday of doo wop groups, since the first doo wop record is generally considered to be The Penguin’s Earth Angel in 1954. There were dozens of similarly named doo-wop groups after that.
For that matter, you could fill a page just with “number” groups in the 50s before the Teddy Bears. Four Aces, The Four Coins, The Four Esquires, The Four Freshmen, The Four Lads, The Four Preps, The Four Voices, The Five Keys, The Three Chuckles, and The Six-Teens [!] all had Top 40 charting hits before 1958. I use them as prime examples because so many earlier bands had numbers in their names.
There is never a time in pop music in which band names don’t exist, as far as I can tell. And of course pop music starts in the early 1800s.
Flipping through Popular Song: Soundtrack of the Century, by Alan Lewens, I find band names of every type and variety of music on virtually every page.
1902’s “Sweet Adeline” was first sung by the Quaker City Four. Sam Cooke came from a gospel group called The Soul Stirrers Black Gospel Quartet. The Ink Spots were popular in the 1930s. Jelly Roll Morton recorded with His Red Hot Peppers. In country music you had The Tennessee Crackerjacks and the Sons of the Pioneers.
Going back into the 19th century, it seems that traveling theatrical performances that we would recognize as musical groups started around 1830 with minstrel shows. These evolved into traveling players, vaudeville, burlesque (not strip tease until well into the 20th century) and a variety of musical types from folk to opera. A good history is With Amusement for All: A History of American Popular Culture Since 1830, by LeRoy Ashby, but it’s too fat and dense to skim at this hour.
I don’t think there is any real answer to the OP’s question. Not matter how far back you go, bands can be found with band names. And any attempt at trying to pin down individual genres is hopeless because no two people ever agree on when music genres started.
A bunch of people get together to make music. They give themselves a name. It’s as old as time.