See this answer from ESRI stating kriging considered exact and this nice description from expert course material that also goes the same route.
Generally, kriging is associated with exactness but according to ESRI:
When semivariogram and covariance models have a nugget effect there is potential for a discontinuity in the predicted surface at the sample data locations.
Although you will see the nugget effect causing inexactness again in actual usage in this hydrology example.
Some authors say it is both! a sound textbook
So pick your answer really. My answer would be that exactness is a vague term (in GIS anyway) that likely should not be used in terms of geostatistical properties. Another question you may want to ask: Is Kriging even an interpolator?