### Godot version

v4.2.dev.custom_build [0ca854232]

### System information

Pop Os 22.04 lts 12th Gen Intel® Core™ i7-12700H × 20 NVIDIA RTX A2000 8GB 32.0GB  ram

### Issue description

A node that has a script attached to it and it has a couple of child nodes (type of a child node doesn't matter). In the script you have an export variable of type Node. You save the scene into a file and you add it in level scene. When you start to duplicate that scene in the level and you'll hover over exported variable to show a node path, it will show an incorrect path. 


https://github.com/godotengine/godot/assets/33091666/75088d1b-c014-41e0-bae3-46610eccbad1



### Steps to reproduce

1. create a scene with a node and a child node.
2. attach a script to the parent node and create an export variable of type Node
3. assign child node to that exported variable through the editor
4. save the scene and create a level scene where you are going to add just saved scene.
5. _Possibly requires a reboot of the engine before seeing the bug_
6. start duplicating nodes in the level scene

### Minimal reproduction project

[NodeTest.zip](https://github.com/godotengine/godot/files/12781547/NodeTest.zip)

return data.name;
}
void Node::_set_name_nocheck(const StringName &p_name) {
	data.name = p_name;
}
void Node::set_name(const String &p_name) {
	ERR_FAIL_COND_MSG(data.inside_tree && !Thread::is_main_thread(), "Changing the name to nodes inside the SceneTree is only allowed from the main thread. Use `set_name.call_deferred(new_name)`.");
	String name = p_name.validate_node_name();
	ERR_FAIL_COND(name.is_empty());
	if (data.unique_name_in_owner && data.owner) {
		_release_unique_name_in_owner();
	}
	String old_name = data.name;
	data.name = name;
	if (data.parent) {
		data.parent->_validate_child_name(this, true);
		bool success = data.parent->data.children.replace_key(old_name, data.name);
		ERR_FAIL_COND_MSG(!success, "Renaming child in hashtable failed, this is a bug.");
	}
	if (data.unique_name_in_owner && data.owner) {
		_acquire_unique_name_in_owner();
	}
	propagate_notification(NOTIFICATION_PATH_RENAMED);
	if (is_inside_tree()) {
		emit_signal(SNAME("renamed"));
		get_tree()->node_renamed(this);
		get_tree()->tree_changed();
	}
}
String Node::get_description() const {