Is whisker layout in a dog a constant?

I've looked around in scientific publications and found no indication that whisker patterns are used anywhere to identify canines (neither lost dogs nor wild animals or those living in zoos).

There are a few research teams and companies who are developing facial recognition algorithms for dogs, but those are all based on machine learning / AI (source 1, source 2, source 3). The problem with these machine learning systems is that no-one can say how exactly it works because the software creates an obscure binary "mask" during the "training" phase that cannot be translated back to individual areas of the picture. You also usually need a lot of pictures of the individual to train the AI to identify the same individual in an unknown image.

The only body feature of dogs that is treated as biometrical identification is the muzzle print of a dog, which is as unique as a fingerprint of a human.

On the other hand, whiskers are not much more than oversized hairs. Hairs grow from follicles, which don't change places in the skin (AFAIK). Whisker spots are an accepted way to identify cats and feline predators and the technique was also applied to polar bears and Australian sea lions. The accuracy of this method strongly depends on the size of population. The sea lion study calculated that the accuracy (read: uniqueness of the pattern) is 98% in a population of 100 animals and drops down to 88% in a population of 1000 animals. Now imagine how many dogs there are in a 260 km radius...

I tried comparing the 2 photos based on what I read about whisker spots and my personal, not scientific opinion is that these are 2 different dogs. As suggested in the comments, you should either ask for more pictures or a video call to see the dog for yourself.