That seems unusual. Those signals should be blocked. It sounds like you have the correct setup with Filter set to Stop. I just tested it out. This is my scene:

and this is the entire .tscn file:
[gd_scene load_steps=6 format=3 uid="uid://dxukvek8fqn5q"]
[ext_resource type="Texture2D" uid="uid://bqori034vw85r" path="res://icon.svg" id="2_tdqg2"]
[sub_resource type="GDScript" id="GDScript_g8evc"]
script/source = "extends Area2D
func on_mouse_enter() -> void:
print(on_mouse_enter)
func on_mouse_exit() -> void:
print(on_mouse_exit)
func _mouse_enter() -> void:
print(_mouse_enter)
func _mouse_exit() -> void:
print(_mouse_exit)
func _input_event(viewport: Viewport, event: InputEvent, shape_idx: int) -> void:
print(_input_event)
"
[sub_resource type="Gradient" id="Gradient_pq4ix"]
[sub_resource type="GradientTexture2D" id="GradientTexture2D_55a8y"]
gradient = SubResource("Gradient_pq4ix")
[sub_resource type="RectangleShape2D" id="RectangleShape2D_ogng8"]
size = Vector2(63, 64)
[node name="Node2D" type="Node2D"]
[node name="Area2D" type="Area2D" parent="."]
position = Vector2(263, 167)
script = SubResource("GDScript_g8evc")
[node name="Sprite2D" type="Sprite2D" parent="Area2D"]
texture = SubResource("GradientTexture2D_55a8y")
[node name="CollisionShape2D" type="CollisionShape2D" parent="Area2D"]
shape = SubResource("RectangleShape2D_ogng8")
[node name="TextureRect" type="TextureRect" parent="."]
offset_left = 205.0
offset_top = 106.0
offset_right = 333.0
offset_bottom = 234.0
mouse_filter = 0
texture = ExtResource("2_tdqg2")
[connection signal="mouse_entered" from="Area2D" to="Area2D" method="on_mouse_enter"]
[connection signal="mouse_exited" from="Area2D" to="Area2D" method="on_mouse_exit"]
Aside from icon.svg it's entirely self-contained. When I move the mouse across the Area2D, I don't get ANY of those print statements. Then if I make the TextureRect invisible and move my mouse across the Area2D, I get all of them.
Either something is weird with your setup or it is a bug in your Godot version. What version does it show when you go to Help/About Godot?