Don't you know who the Monster was???
It was "Nelson - Louise's Fiancee in Framed Photograph"!!!
The picture on the nightstand was of Louise standing arm-in-arm with a tall, smiling man... actually it was director Roger Corman in an uncredited role.
I hadn't seen the film for decades but when I did, and thought about it a little more that I probably should, I realized who the Monster's "character" was supposed to be: missing boyfriend "Nelson" that is mentioned very briefly right at the beginning of the film!
Consider these points:
The thing came straight to the house, as if knowing it was there and that Louise would also be there. It called to her (telepathically) and ONLY her. It used her NAME (again, telepathically). The clincher was when Louise contemplated the photograph at the end of the film and the cut-away back to the dead Monster.
So... there you have it!