Yes, this database is called "Math Genealogy", genealogy.math.ndsu.nodak.edu.
Its value and reliability has been much discussed, but on my opinion, it is useful
and entertaining. AMS is probably of the same opinion: it includes links to it
Some points of criticism are the following: the notions of PhD degree and PhD adviser is relatively recent; it did not exist in all countries at all times, though the database covers
some people back to 15 century.
Until 18 century mathematics was not a profession, so inevitably the database contains many people who cannot be called mathematicians, or mathematicians who
earned a degree for other things.
The data about 19-20 centuries PhD does not cover various countries uniformly.
With all these drawbacks, one still can learn something interesting from this database.
BTW it is not surprising that well known mathematicians come from well known advisers:-) Exceptions from this rule are more surprising.