### 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.

return nodes.has(p_name);
}
Vector<StringName> AnimationNodeBlendTree::get_node_connection_array(const StringName &p_name) const {
	ERR_FAIL_COND_V(!nodes.has(p_name), Vector<StringName>());
	return nodes[p_name].connections;
}
void AnimationNodeBlendTree::remove_node(const StringName &p_name) {
	ERR_FAIL_COND(!nodes.has(p_name));
	ERR_FAIL_COND(p_name == SceneStringNames::get_singleton()->output); 
	{
		Ref<AnimationNode> node = nodes[p_name].node;
		node->disconnect("tree_changed", callable_mp(this, &AnimationNodeBlendTree::_tree_changed));
		node->disconnect("animation_node_renamed", callable_mp(this, &AnimationNodeBlendTree::_animation_node_renamed));
		node->disconnect("animation_node_removed", callable_mp(this, &AnimationNodeBlendTree::_animation_node_removed));
		node->disconnect_changed(callable_mp(this, &AnimationNodeBlendTree::_node_changed));
	}
	nodes.erase(p_name);
	
	for (KeyValue<StringName, Node> &E : nodes) {
		for (int i = 0; i < E.value.connections.size(); i++) {
			if (E.value.connections[i] == p_name) {
				E.value.connections.write[i] = StringName();
			}
		}
	}
	emit_signal(SNAME("animation_node_removed"), get_instance_id(), p_name);
	emit_changed();
	emit_signal(SNAME("tree_changed"));
}
void AnimationNodeBlendTree::rename_node(const StringName &p_name, const StringName &p_new_name) {