CLR Assembly C# DBMS

You couldn't just make a DBMS use the .NET CLR but if the DBMS supports a way to call arbitrary code via standard entry points then you can create a bridge. Oracle for example has the OCI (Oracle Call Interface) that you could use to bridge to .NET assemblies.

You cannot use the CLR type you register in SQL server database directly in an Oracle database (and I guess the same holds true for MySql and other databases). Each database that supports User defined datatypes and has a .NET provider usually has a framework in place for representing the UDT in the the CLR world.

For example, in Oracle you need to implement the 'Oracle.DataAcess.Types.IOracleCustomType' interface to be able to convert between the UDT and CLR type and the 'IOracleCustomTypeFactory' to create instances of the custorm type. You can find more information here.