First, probiotics are specific to their environment: what works in the intestines doesn't necessarily work for the skin's microbiome.
Microbiome describes the community of microorganisms - such as bacteria, fungi, and viruses - that inhabit a particular environment, e.g. your intestines. Until scientific/medical community cracks the code on what works for the skin's microbiome, it's more of an optimistic guessing game and vulnerable to marketing overstatements.
Second, probiotics are live microorganisms (like bacteria & yeast) that when introduced in adequate amounts convey a health benefit, for example by working within the intestinal flora to balance that microbiome. Even if we knew exactly which probiotics were optimal for the skin, it seems doubtful that commercial pet shampoos would be able to sustain enough of these live microorganisms to benefit the skin especially since many shampoos are not in contact with the dog's skin for longer than 15-20 minutes. (You should see my dogs shoot out of the tub, flinging suds and water everywhere!)
Probiotics can definitely have a place in your dog's overall care when introduced into their diet. Talk to your vet about whether your dog's gut needs rebalancing to calm inflammation that shows up as itchy, inflamed skin or to address gastrointestinal upset like chronic constipation or diarrhea.
If you know of research that shows topical probiotics specifically support skin health, please let me know. Until then, I've got it filed under "marketing not science" but am hopeful this will change.