I'll admit I didn't think of how unfeasible the idea is, but the most likely answer is that theirs so many escaped slaves and catching them can be so problematic that theirs a entire guild of freelance government sponsored slave catchers that operate both inside and outside the countryStill doesn’t make any sense whatsoever.
Why would there be a registry of birthmarks? Apparently this is a family thing, so are there really that many slaves with birthmarks that setting up a registry is warranted?
Again, how many resources does this nation allot to recapturing slaves? We have independent, roving bands of slave catchers that will put out elaborate charades to catch a single suspected slave. They are equipped with magical items that allow them to detect whether some one is a slave or not (how does that even work? What happens if the law changes? Does the item stop working?). Now we have a registry of slave birthmarks that is apparently easily accessible by this 7-man strong band of roving slavers.
Theirs also the fact that the countries police force has a entire division devoted to catching all escaped slaves, and their children, that are still inside the country and the secret service has a entire division devoted to catching all escaped slaves, and their children that manage to get out of the country