I may be wrong but to me it looks like matted hair, saliva, and dried food or dirt mixed together and stuck on your dog's lip. Try picking it off. My dog used to get those a lot. Some dogs just need their mouths wiped from time to time. Just peel it off.
Yeah...maybe...your way is the more cautious and better way. But the first time. I just saw lint/hair all matted and stuck on his lip and I didn't think anything of it...just peeled it right off. And from then on, it was just an easy peel off....no inflammation.My guy had lips that looked like this often. The lips were loose at the corners and when wet with saliva, would pick up hair and fabric fibres. This needs to be soaked off, not pulled. There could be underlying inflammation that causes tissues to weep and become infected, so gently remove it and if there is raw skin, time to see the vet.