### 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.

String cur_transition_request = get_parameter(transition_request);
	int cur_current_index = get_parameter(current_index);
	int cur_prev_index = get_parameter(prev_index);
	double cur_time = get_parameter(time);
	double cur_prev_xfading = get_parameter(prev_xfading);
	bool switched = false;
	bool restart = false;
	bool clear_remaining_fade = false;
	if (pending_update) {
		if (cur_current_index < 0 || cur_current_index >= get_input_count()) {
			set_parameter(prev_index, -1);
			if (get_input_count() > 0) {
				set_parameter(current_index, 0);
				set_parameter(current_state, get_input_name(0));
			} else {
				set_parameter(current_index, -1);
				set_parameter(current_state, StringName());
			}
		} else {
			set_parameter(current_state, get_input_name(cur_current_index));
		}
		pending_update = false;
	}
	if (p_time == 0 && p_seek && !p_is_external_seeking) {
		clear_remaining_fade = true; 
	}
	if (!cur_transition_request.is_empty()) {
		int new_idx = find_input(cur_transition_request);