The model starts from 64x64 8bit RGB image of noise (random pixels) so technically 1 in 3_145_728 (64x64x256x3) but most will probably be very close to each other as the color difference won't be that much. The image is then further upsampled by two other models which will change some details, but shouldn't affect the general composition of an image.
Maybe I'm wrong, but with these diffusion models there is randomness in every sampling step too not just in the initialization and they can have 1000 steps to generate a single image.
Ah good point, this would introduce more variation if the initial noise is close, but if the initial noise is exactly the same it probably means it was initialized with the same seed and the rest of the generation will be the same since the random algorithms are deterministic.
In other words, if another person needed a logo and used the same phrase how long on average until they get a duplicate of your image?