### Godot version

v4.1.1.stable.official [bd6af8e0e]

### System information

Godot v4.1.1.stable - Windows 10.0.19045 - Vulkan (Forward+) - dedicated NVIDIA GeForce GTX 1060 6GB (NVIDIA; 31.0.15.3179) - Intel(R) Core(TM) i5-8500 CPU @ 3.00GHz (6 Threads)

### Issue description

When connecting the `animation_started` signal from an animation tree, the connected method is always called twice.
This only occurs with the `animation_started` signal, the `animation_finished` signal is only emitted once.

This unfortunately makes relying on the signal impossible.

### Steps to reproduce

(See minimal production project)
- Use `animation_started.connect()`
- Print something out in the connected method (the example project prints the `anim_name` parameter)
- method is called twice -> two prints in the console

### Minimal reproduction project

[anim_tree_signal_bug.zip](https://github.com/godotengine/godot/files/12229864/anim_tree_signal_bug.zip)

The animation tree travels to the 'move' animation when `ui_accept` is pressed (see `anim_tree.gd`) and has print statements inside the methods connected to both the `animation_started` and `animation_finished` signal.

List<StringName> names;
	for (const KeyValue<StringName, Node> &E : nodes) {
		names.push_back(E.key);
	}
	for (const StringName &E : names) {
		String prop_name = E;
		if (prop_name != "output") {
			p_list->push_back(PropertyInfo(Variant::OBJECT, "nodes/" + prop_name + "/node", PROPERTY_HINT_RESOURCE_TYPE, "AnimationNode", PROPERTY_USAGE_NO_EDITOR));
		}
		p_list->push_back(PropertyInfo(Variant::VECTOR2, "nodes/" + prop_name + "/position", PROPERTY_HINT_NONE, "", PROPERTY_USAGE_NO_EDITOR));
	}
	p_list->push_back(PropertyInfo(Variant::ARRAY, "node_connections", PROPERTY_HINT_NONE, "", PROPERTY_USAGE_NO_EDITOR));
}
void AnimationNodeBlendTree::_tree_changed() {
	AnimationRootNode::_tree_changed();
}
void AnimationNodeBlendTree::_animation_node_renamed(const ObjectID &p_oid, const String &p_old_name, const String &p_new_name) {
	AnimationRootNode::_animation_node_renamed(p_oid, p_old_name, p_new_name);
}
void AnimationNodeBlendTree::_animation_node_removed(const ObjectID &p_oid, const StringName &p_node) {
	AnimationRootNode::_animation_node_removed(p_oid, p_node);
}
void AnimationNodeBlendTree::reset_state() {
	graph_offset = Vector2();
	nodes.clear();
	_initialize_node_tree();
	emit_changed();
	emit_signal(SNAME("tree_changed"));
}
void AnimationNodeBlendTree::_node_changed(const StringName &p_node) {