Mugwort (Artemisia vulgaris), Common Sage (Salvia officinalis), White Sage (Salvia apiana) and Prairie Sage (Salvia azurea) have a little in common. They're all medicinal plants, and the three sages belong to the same genus and family. However, Mugwort is in the Sunflower family, while the other three are in the Mint family.
Both Mugwort and Common Sage contain some amount of the chemical Thujone, which is a stimulant and nerve agent. If you're allergic to it, I wouldn't chance an encounter with White Sage, although it has a very low Thujone content. They may have a few minor similarities in chemical composition, but I don't think there's a chance of cross allergic reactions between Mugwort and White Sage.
Hello and hope you are doing well.
As they belong to the same family there could be cross allergic reactions.
Hope this helped and do keep us posted.