In many cases, each company has its own database (or several
databases), so I would estimate that there are about as many
databases as there are companies.
If you are referring to the DBMS - the software that manages the
databases - you can see a list of the most common ones in the
following Wikipedia articles:
* Comparison of relational database management systems
* Comparison of object database management systems
* Comparison of object-relational database management
systems