Barbara and Hu Tao have never canonically interacted yet in the canon timeline, as Barbara and Hu Tao are from different nations being which Mondstadt and Liyue. Both also do not have that much similarities, but the two are playable characters. But they have a lot of differences, as already said, Barbara is from Mondstadt while Hu Tao is from Liyue, their personalities are very different. Hu Tao is a prankster, most of the Liyue characters seems to have a bad relationship with her. And then Barbara on the other hand, is a cheerful and nice girl who sings. Having good relationships with all of the Mondstadt characters.
Both have composed songs, Barbara is an idol in Mondstadt, even in one of her idle animations she sings. Meaning that she has already composed a couple of songs. And then for Hu Tao, she has composed a song called, "Hilitune". The two work at somewhere, Barbara works as a deaconess in the Cathedral. And Hu Tao works at Wangsheng Funeral Parlor, in which, she's also the 77th owner of it.
The ship was created on and is mainly popular on Twitter, though BarbTao wasn't as popular among most other platforms. BarbTao gained some attention, because of the artist named, Bunsketches on Twitter has drawn an image of the two together, which gotten famous as the tweet has 16322 likes and 4062 retweets. This also goes the same to Jeandou. As the Characters never interacted, the ship is almost nonexistent among the Genshin Impact fandom, although it has some fans nonetheless.
While BarbTao gained some attention from Bunsketches, a tweet then came on Twitter and have gotten popular, the image depicts Hu Tao being Hanako while Barbara being Yashiro Nene. Many fan arts were now drawn due to both of these tweets. As said, Barbara and Nene does have some similarities as both are friendly, and trusting and is dedicated to keeping the ones she cares about safe, though both, Barbara and Nene may not look exactly the same. Both, Hu Tao and Hanako have quite the similarities. As both of their appearances looks the same if you compare it, their personalities tends to be mischievous, both also enjoys teasing others.
This also serves the Opposites Attract trope, it's a popular trope among the ships. Being that, BarbTao is also a rare pair among the ships in the Genshin Impact fandom. As it may not be popular, they were being shipped due to their aesthetics and personality (even their dynamics). And some also like the small fan content, such as fan arts, that exists for BarbTao. It was also the reason why how both, Barbara and Hu Tao became a pairing. BarbTao's rival ships include Rosabara, Barbelle, Barbennett, Taoling, HuXiao and YanTao. Which are some of both, Barbara's and Hu Tao's popular ships. There are currently 15+ works depicting the two romantically on AO3.