I think you've greatly misunderstood how idioms work. The phrase doesnt "literally indicate the listener is a dog". The origin of the idiom comes from behaviour of hunting dogs, but it's an idiom. To take the first line from Wikipedia on idioms "An idiom is a phrase or expression that typically presents a figurative, non-literal meaning attached to the phrase." (Emphasis mine)

@JMac The keyword is "barking". Humans don't bark. Dogs do bark. So, it literally indicates the listener is a dog. You don't need to tell me it's just an idiom. In some non-English speaking cultures, saying someone is barking is offensive.

Suggesting the phrase is "plainly offensive in any culture" is a wild overstatement, and immediately contradicted by your observation that "native English speakers find this idiom inoffensive".

@Nobody I think I do need to tell you it's an idiom, because you still seem to have missed the point about them. The word "bark" is part of the idiom, its idiomatic, and therefore doesn't indicate the listener is a dog, or barking, just like it doesnt indicate there is actually a tree involved. I think it's fair to say idioms can be tricky with non-native speakers, but a misunderstanding of idioms doesnt mean anyone is being literally accused of being a dog, barking, or anything to do with a real tree.

@JMac We are talking in English because this is an English only platform. This is the same situation between the OP and their colleagues. However, it does not mean we don't need to respect those non-English speaking cultures. If the listener does think it's offensive, it's offensive to that person. Nothing to do with the idiom. (The OP was not teaching their colleagues an English lesson).

@Nobody Plenty of other answers touch on that in a much clearer way than this answer. Especially when they add in terms like "too stupid" when even a literal reading of the idiom just shows the dog is mistaken, not necessarily "stupid".

@JMac You don't need to teach me English. I lived in the US for over 30 years. However, I am not a native English speaker. From the culture I am from, saying someone is barking is disrespectful. There is no need for the OP to make an enemy.

Apart from all the nitpicking, this seems to be the best answer. It unpacks the meaning of the idiom for someone who doesn't necessarily know the idiom. It seems entirely possible that a first-time hearer of this idiom would take umbrage at being called---even in a roundabout way---a stupid dog.

@JMac I hate to ask, but should I explain the difference between literally and figuratively?

@NuclearHoagie If I called you a dog who's too stupid to find it's bird, you would be entirely reasonable in saying I attempted a slight.

@user121330 I would, but the words in an idiom do not take their literal dictionary definitions. "You're barking up the wrong tree" absolutely does not suggest in any way that you are making sounds like a dog at woody plant. "You are a stupid dog", on the other hand, isn't an idiom, and carries exactly the meaning of its individual constituent words.

@user121330 If you're going to use "literally" in a non-figurative sense, then all it indicates is that the listener is "literally barking up a tree which is not the correct one". As a completely literal phrase, it means very little. Even "barking up" is idiomatic on it's own. Your "literal" definition requires as much or more knowledge external to the wording than the idiomatic definition, which makes it pretty non-literal to me.

@JMac The distinctions you are drawing probably have value. My questions for you are whether you find it plausible, based on the text, the origin and what we know of the context of its usage, that the listener took offense at this idiom? Does it serve anything other than the speaker's ego to defend its usage? How many times would this instance or something similar need to occur before it becomes harassment?

Your assertion that it "literally indicates that the listener is a dog" is blatantly false. The wiki you linked to says "The phrase is an allusion to" which places it firmly at odds with your assessment.

The first two sentences make this a very poor answer. This idiom absolutely DOES NOT mean the listener is a stupid dog. The rest of the answer is useful. Please strike the obnoxious first sentences.