### 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 (!nodes.has(p_output_node) || p_output_node == SceneStringNames::get_singleton()->output) {
		return CONNECTION_ERROR_NO_OUTPUT;
	}
	if (!nodes.has(p_input_node)) {
		return CONNECTION_ERROR_NO_INPUT;
	}
	if (p_input_node == p_output_node) {
		return CONNECTION_ERROR_SAME_NODE;
	}
	Ref<AnimationNode> input = nodes[p_input_node].node;
	if (p_input_index < 0 || p_input_index >= nodes[p_input_node].connections.size()) {
		return CONNECTION_ERROR_NO_INPUT_INDEX;
	}
	if (nodes[p_input_node].connections[p_input_index] != StringName()) {
		return CONNECTION_ERROR_CONNECTION_EXISTS;
	}
	for (const KeyValue<StringName, Node> &E : nodes) {
		for (int i = 0; i < E.value.connections.size(); i++) {
			const StringName output = E.value.connections[i];
			if (output == p_output_node) {
				return CONNECTION_ERROR_CONNECTION_EXISTS;
			}
		}
	}
	return CONNECTION_OK;
}
void AnimationNodeBlendTree::get_node_connections(List<NodeConnection> *r_connections) const {