A database is an organized collection of data. This web site is largely our public database.
MySQL is by far the best database server. It does its job well, and it's open source.
Then comes the interfacing of SQL databases with, for instance, this website. Unless you wish to code everything yourself or enter the data at a low (and rather rebutting) level, you want another layer of software to help you create, read, update and delete entries, operations collectively known as CRUDing.
There is such things as WikiDB which in great in spirit but not sufficiently advanced.
No doing it at the level of the mediawiki engine, there are things like DaDaBIK, which used to be free but turned commercial (the GPL older version is still available on the web), so we cannot recommend it.
A great is client to deal with SQL databases is adminer.
Possibly the SQL data can be retrieved from a wiki through extensions such as SQL2Wiki.