In the non-AI world, a database server mostly always just executes any query you give it to, assuming right permissions.

They are not responsible only in the way they wouldn't be responsible for an application-level sql injection vulnerability.

But that's not to say that they wouldn't be capable of adding safeguards on their end, not even on their MCP layer. Adding policies and narrowing access to whatever comes through MCP to the server and so on would be more assuring measures than what their comment here suggest around more prompting.