### 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.

ERR_FAIL_COND(!nodes.has(p_name));
	ERR_FAIL_COND(nodes.has(p_new_name));
	ERR_FAIL_COND(p_name == SceneStringNames::get_singleton()->output);
	ERR_FAIL_COND(p_new_name == SceneStringNames::get_singleton()->output);
	nodes[p_name].node->disconnect_changed(callable_mp(this, &AnimationNodeBlendTree::_node_changed));
	nodes[p_new_name] = nodes[p_name];
	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] = p_new_name;
			}
		}
	}
	
	nodes[p_new_name].node->connect_changed(callable_mp(this, &AnimationNodeBlendTree::_node_changed).bind(p_new_name), CONNECT_REFERENCE_COUNTED);
	emit_signal(SNAME("animation_node_renamed"), get_instance_id(), p_name, p_new_name);
	emit_signal(SNAME("tree_changed"));
}
void AnimationNodeBlendTree::connect_node(const StringName &p_input_node, int p_input_index, const StringName &p_output_node) {