### 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.

if (new_idx >= 0) {
			if (cur_current_index == new_idx) {
				if (allow_transition_to_self) {
					
					restart = input_data[cur_current_index].reset;
					clear_remaining_fade = true;
				}
			} else {
				switched = true;
				cur_prev_index = cur_current_index;
				set_parameter(prev_index, cur_current_index);
				cur_current_index = new_idx;
				set_parameter(current_index, cur_current_index);
				set_parameter(current_state, cur_transition_request);
			}
		} else {
			ERR_PRINT("No such input: '" + cur_transition_request + "'");
		}
		cur_transition_request = String();
		set_parameter(transition_request, cur_transition_request);
	}
	if (clear_remaining_fade) {
		cur_prev_xfading = 0;
		set_parameter(prev_xfading, 0);
		cur_prev_index = -1;
		set_parameter(prev_index, -1);
	}
	
	if (restart) {
		set_parameter(time, 0);