### 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_INDEX_V(p_input, get_input_count(), false);
	return input_data[p_input].auto_advance;
}
void AnimationNodeTransition::set_input_reset(int p_input, bool p_enable) {
	ERR_FAIL_INDEX(p_input, get_input_count());
	input_data.write[p_input].reset = p_enable;
}
bool AnimationNodeTransition::is_input_reset(int p_input) const {
	ERR_FAIL_INDEX_V(p_input, get_input_count(), true);
	return input_data[p_input].reset;
}
void AnimationNodeTransition::set_xfade_time(double p_fade) {
	xfade_time = p_fade;
}
double AnimationNodeTransition::get_xfade_time() const {
	return xfade_time;
}
void AnimationNodeTransition::set_xfade_curve(const Ref<Curve> &p_curve) {
	xfade_curve = p_curve;
}
Ref<Curve> AnimationNodeTransition::get_xfade_curve() const {
	return xfade_curve;
}
void AnimationNodeTransition::set_allow_transition_to_self(bool p_enable) {
	allow_transition_to_self = p_enable;
}
bool AnimationNodeTransition::is_allow_transition_to_self() const {
	return allow_transition_to_self;
}
double AnimationNodeTransition::_process(double p_time, bool p_seek, bool p_is_external_seeking, bool p_test_only) {