Danyang is the Japanese pronunciation for her new slave name.
"Tan Yang" is Wade-Giles which was the standard used at the time (and was still the standard until recently in the ROC). "Danyang" is the same thing in Pinyin, which is the current standard. The guys at pixiv are actually using a phonetic approximation from Mandarin (Tan'yan, タンヤン) instead of the on'yomi for the characters (Tan'you, タンヨウ).