That's an interesting take on the idea of Dan Heng's tail, because I always interpreted it that it very much is his tail (because he specifically told Luocha that some Vidyadhara have more draconic traits after going through unspecified trials namely to become High Elders though Hoyo's lore is very vague on specifics, but if Dan Heng inherited the horns from Dan Feng there's no reason he wouldn't have gotten other draconic traits) and he uses cloudhymn/shapeshifting to hide it, like he does the rest of his appearance just on a smaller scale. It manifests for the short time that he's in his meditation state and more connected to his power, but then he hides it again so that it can't be used as a target. Given the Preceptors use Bailu's tail to block some of her power it wouldn't be strange if he's afraid of them doing the same to him, or if that's something they used on him in the Shackling Prison to control him. That was how I interpreted it, at least, and I assume it's a common reasoning for why people draw him with it, being a sign he feels safe and comfortable enough to stop hiding it.
Just about everything stated about Bailu should be treated with skepticism, including anything Bailu has assumed of herself, because she is the product of illegal experiments the Vidyadhara Preceptors conducted with Disciples of Sanctus Medicus.
This plotline has yet to be fully resolved and the conspiracy is an ongoing plot point. As a result, explicit confirmation is lacking and analysis relies heavily on inferences.
In the "Undestroyed Notes" readable, it is noted that Bailu's creation was an accidental "miracle" in a "Bloomborn Scion" project. This project (which is ongoing) is probably part of the Preceptors' efforts to facilitate vidyadhara reproduction (their foremost concern, after their successful coup against Dan Feng).
The dragonhorn pillory Bailu wears was specifically created for her by an attendant: no such device is mentioned in recountings of Dan Feng's arrest or imprisonment and Dan Heng does not note one either. This is either a plot hole (a glaring oversight by the Shackling Prison and some very sadistic Preceptors) or it's supporting evidence that vidyadhara do not naturally have tails.
Bailu's tail is an extreme abnormality for a vidyadhara and foreshadows that she is not only a fraudulent high elder (biologically speaking) but that she may not be a vidyadhara at all.
The "Sanctus Medicus Devils Criminal Evidence Records" readable, which documents the Order's experiments with vidyadhara bone marrow, at least one subject is noted to grow an extra limb and most developed physical traits from other species.
While these features did include such things as scales and horns, they were not noted to be those of a vidyadhara. It is very probable that not even Bailu's horns are truly that of a "vidyadhara high elder" and are, instead, just another mutation.
(culturally speaking, Dan Heng himself has judged Bailu "worthy" of her title and the regular vidyadhara of the Luofu do too: theirs are the only opinions that truly matter, narratively speaking)
Bailu having a tail while Dan Heng does not is a plot point. A subtle one, perhaps, but such is the nature of slowburn political conspiracies. Dan Heng's "tail" is, therefore, just another cloudhymn construct. That said, it's still unclear how "real" such constructs may be: is the dragon sapient? is the tail? are they based on an actual dragon, the one mentioned in tales of the Sedition and from whom Bailu was supposedly constructed?
The fun of these mysteries is lost when Bailu The Full-Time Red Herring is taken at face value. The ambiguity is part writing choice, part misleading translations, and partly poor implementation of lore into casual gameplay. Fan artists popularized this misconception and tailed Dan Heng is, indeed, very cute but it's 100% fanon.