QR codes also encode text, just like UPC-A. As far as I'm aware, no barcode reader can actually interpret the letters at the bottom of the code. They're just there for a human failover when the barcode gets messed up.
With that in mind, there's nothing preventing you from putting the encoded text of a QR code below it. I have a QR code in my living room that guests can scan to connect to the Wi-Fi, and I put the SSID and password below the QR code on the same page. It's not the exact encoded string, but it serves the same purpose.
My first "real" job was working for an industrial automation company, where I did a lot of work on RFID and barcode readers. Barcodes (both 2D and 3D) are delightful things. There's an enormous variety out there, and you would never know unless you look closely, because they're so well-supported. I'd bet the barcode reader at your local grocery store can read at least 9 different encodings. Some are probably physically capable of reading QR (and other 3D codes like Aztec), but the firmware has been instructed to ignore them.
In the industry, we called "normal" barcodes "2D," and stuff like QR and Aztec were called "3D" barcodes. Think of the "Z" axis as the coloration of a given point.
In all barcodes I'm aware of, that Z axis only has one bit of resolution (i.e. black or white, 1 or 0). If you had an encoding with more colors, you'd be increasing the resolution of that axis, but you wouldn't be adding a new axis per se.
Honestly, I also think 1D/2D would make more sense. But, 2D/3D is the nomenclature I saw being used among industry professionals on both the marketing and engineering sides.
With that in mind, there's nothing preventing you from putting the encoded text of a QR code below it. I have a QR code in my living room that guests can scan to connect to the Wi-Fi, and I put the SSID and password below the QR code on the same page. It's not the exact encoded string, but it serves the same purpose.
My first "real" job was working for an industrial automation company, where I did a lot of work on RFID and barcode readers. Barcodes (both 2D and 3D) are delightful things. There's an enormous variety out there, and you would never know unless you look closely, because they're so well-supported. I'd bet the barcode reader at your local grocery store can read at least 9 different encodings. Some are probably physically capable of reading QR (and other 3D codes like Aztec), but the firmware has been instructed to ignore them.