My dog is a female Labrador, we have had her for around 6 years. Today I was walking her and as we were passing another dog, she went to the butt so I thought she was going to smell it - but 3 seconds after, they both started barking. I said "sorry" and walked in a different direction. My dog had that dog's hair in her mouth, now I feel really guilty and was wondering why she might have done that and if I should get my mum to take her to the vet.
If everything else is OK/normal, no. There are far too many possible reasons for this.
Most likely the other dog is changing coat right now and it was just coincidence of having your dog "pull" that hair out.
If there's been some other factor involved, it's likely been the other dog in some way. For example, one of our Huskies really dislikes ticks. So if it smells one on the ground, in its own fur, or in another dog's fur it'll actively try to "hunt" and kill it. Other dogs (and owners!) might misinterpret such behavior as being aggressive.