The line "I wouldn't want any more marks on my cat license" is a reference to the Monty Python "Fish license" sketch.
Doronbol's chat remarks and examine option infers that she is a dog in disguise. The "makes me want to wag my tail" chat comment definitely further implies her being a dog, as cats only wag their tails when annoyed and use it as a warning before they attack. This would not explain why you need a catspeak amulet to talk to her, though. Possibly she was a dog that was somehow turned into, or reincarnated as, a cat. She might even be pretending to not understand you while not wielding a catspeak amulet.
The name Doronbol is similar to the Hungarian word "dorombol", which means "purring".
When the player attempts to use the log chest without having completed the Elite Varrock Diary Doronbol will tell the player to get away from the chest until the player proves him or herself a true ally of Varrock. Doronbol speaks in the common tongue during this even if the player is not wearing a catspeak amulet.