### 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.

r_list->push_back(PropertyInfo(Variant::FLOAT, time, PROPERTY_HINT_NONE, "", PROPERTY_USAGE_NONE));
}
void AnimationNodeAnimation::_validate_property(PropertyInfo &p_property) const {
	if (p_property.name == "animation" && get_editable_animation_list) {
		Vector<String> names = get_editable_animation_list();
		String anims;
		for (int i = 0; i < names.size(); i++) {
			if (i > 0) {
				anims += ",";
			}
			anims += String(names[i]);
		}
		if (!anims.is_empty()) {
			p_property.hint = PROPERTY_HINT_ENUM;
			p_property.hint_string = anims;
		}
	}
}
double AnimationNodeAnimation::_process(double p_time, bool p_seek, bool p_is_external_seeking, bool p_test_only) {
	AnimationPlayer *ap = state->player;
	ERR_FAIL_NULL_V(ap, 0);
	double cur_time = get_parameter(time);
	if (!ap->has_animation(animation)) {
		AnimationNodeBlendTree *tree = Object::cast_to<AnimationNodeBlendTree>(parent);
		if (tree) {
			String node_name = tree->get_node_name(Ref<AnimationNodeAnimation>(this));
			make_invalid(vformat(RTR("On BlendTree node '%s', animation not found: '%s'"), node_name, animation));
		} else {
			make_invalid(vformat(RTR("Animation not found: '%s'"), animation));
		}