First, because eggs can develop into chickens, chickens are meat, so eggs are also meat dishes.
Second, eggs belong to birds' eggs in the menu. Birds are meat dishes, so eggs are meat dishes. They belong to the same category.
Third, the difference between vegetarian food and vegetarian food is to classify plant food as vegetarian food, animal food as vegetarian food, and eggs are of course animal food, so they are vegetarian food. There is also evidence that vegetarian restaurants do not use eggs, so eggs are meat dishes.