Well... I hoped you may be onto something, terskac. Given your thoughts about one device at a time, I powered down my iPad (but just left my MacBook Pro asleep), and then started sending texts from my wife's iPhone. Whaaaat? I started getting notifications on my iPhone repeatedly. So, then I turned the iPad back on and resumed sending me texts. Amazingly, notifications persisted on my iPhone. And they appeared also on my iPad... but inconsistently. Sometimes just popping up, sometimes chiming. Apple just can't seem to figure out how to handle multiple devices.
Unfortunately, even this solution failed later, even with the iPad powered off and after setting my iPhone to not forward messages to my iPad. So, back to TURNING OFF BLUETOOTH to keep notifications coming (my only reliable solution).
Like all of us on this post, I've been fighting this for nearly a year now when my iPhone 11Pro is locked. I was hoping iOS 15 would fix it, but you're saying it doesn't. Blast it, Apple!