Find the mutual information between the old data and the old data + new data points. If the change is too high, they won't belong to the same distribution.
You can also do clustering and find distance from the cluster heads.
Naming linear data or non-linear data is a bit misleading and wrong I would say. Instead, there is linear relation and non-linear relation between variables would be better and correct naming. It can easily be explained by example from real life.
Non-linear relation: Any kind of relationship that is not linear can be put forward as an example (e.g. ...