Is it possible to update Siri on my iPhone 13?

If you update the iOS on your iPhone, it will also update Siri. You should be able to update by selecting Settings > General > Software Update. The links below may be helpful.

Update your iPhone or iPad - Apple Support

If Siri isn't working on your iPhone - Apple Support

You can update Siri but you will not get Apple Intelligence along with the update. Apple Intelligence is available only on iPhone 15 Pro and later if that’s what you are actually asking about.