Thank you for your post. Siri does not have a built-in native capability to write arbitrary content directly to NSPasteboard / UIPasteboard out of the box without going through a Shortcut. However I would suggest you to look into AppIntents like @AssistantIntent or use the NSPasteboard API in a custom AppIntent? Looking at the documentation for the restrictions as I never have done such a thing.
In macOS, NSPasteboard.general is accessible without Siri however, nothing stops you from creating a custom AppIntent that uses that API as far as I know. On iOS and iPadOS, UIPasteboard.general works inside AppIntent.perform() but AppIntents on iOS lack direct clipboard write hooks beyond custom AppKit-style extensions.
Read the documetaton well as there are foreground/background restrictions https://developer.apple.com/documentation/appintents/appintent/supportedmodes
Also pasting raises a user alert that prompts the user for approval before the app gains access to pasteboard contents (UIPasteboard.general.string) https://developer.apple.com/documentation/uikit/uipastecontrol
Albert