How does a Decision Tree handle an unknown category in a split?
There are two perspectives to this question, from a mathematical or machine learning perspective and from a technical perspective. From a technical perspective this depends on the implementation of the decision tree. An unseen value is not that different from a missing value, and sklearn for example does not deal with unknown values well and will fail on unseen or unknown values. Other tree implementations might deal better with this.

From a mathematical or ML perspective, without making some assumptions you cannot solve this problem of course. However an assumption you could make is that unknown or unseen values are average values. With that assumption you could follow both paths on the split where the unknown value occurs. Then you collect both results from the split and weigh them by the number of training samples that happened there. That way you can still make a prediction for these unknown values.