### 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.

set_parameter(time_to_restart, -1);
	} else if (!do_start && !cur_active) {
		if (cur_time_to_restart >= 0.0 && !p_seek) {
			cur_time_to_restart -= p_time;
			if (cur_time_to_restart < 0) {
				do_start = true; 
			}
			set_parameter(time_to_restart, cur_time_to_restart);
		}
		if (!do_start) {
			is_shooting = false;
		}
	}
	bool os_seek = p_seek;
	if (clear_remaining_fade) {
		os_seek = false;
		cur_fade_out_remaining = 0;
		set_parameter(fade_out_remaining, 0);
		if (is_fading_out) {
			is_fading_out = false;
			set_parameter(internal_active, false);
			set_parameter(active, false);
		}
	}
	if (!is_shooting) {
		return blend_input(0, p_time, p_seek, p_is_external_seeking, 1.0, FILTER_IGNORE, sync, p_test_only);
	}
	if (do_start) {
		cur_time = 0;
		os_seek = true;