Why my dog obsessively licks other dog's mouths?

There are various reasons dogs can do this. These include play time, submission, "i'm sorry", and hunger.

For play time they do it to signal to the other dog that they are ready to play. If the other dog is not interested you may want to pull the licking pup away as this may irritate the other.

For submission this shows the pecking order. A dog licking another dogs mouth shows that he is submissive and below the other dog similar to one a dog rolls on their back. Once again, keep an eye on this as the dominant dog might get snappy if irritated.

When saying sorry the dog will gently lick the other dogs face and sometimes lift their paw gently. The other dogs say "its ok" by licking back.

Lastly it can be a sign that they are hungry. This is usually seen in puppies. Their instinct is to lick their mothers face when they are hungry. Use this as a sign of when to feed the pup.

Dogs licking each others' lips is actually a completely normal behavior. It's sometimes a sign of affection or a sign the dog is ready to play, but more than anything it's a behavior that comes from early puppy-hood when puppies would lick their mother's mouth to encourage the mother to regurgitate food for them. You mentioned the safety of your dog being your concern. If you notice that your pup is repeatedly licking another dog, I'd recommend to just pay close attention to the dog being licked. If the other dog is trying to move away, you may want to intervene.

It typically means they are trying To say hello or make peace.

I believe it is a sign that the dog who licks wants to play.

It's a behavior that comes from early puppyhood when puppies used to lick their mothers lips. The behavior was meant to encourage the mother pup to regurgitate food for them.