I do not think it means what you think it means
When someone says, "You eat like a bird!" they usually intend to imply that you are hardly eating anything at all, that you are a light eater, just pecking at your food and taking small bites. But what they're really implying, if you think of the phrase in terns of how birds actually eat, is the exact opposite. While it looks to us that birds take teensy bites of food here and there, over the course of the day they eat eat loads of food of food!
Because birds, especially smaller birds, do everything at a speedy pace — from wing beats to heartbeats — they need a lot of food to keep fueled for such rapid activity. That means that many bird species will eat between 1/4 to 1/2 of their body weight every day. If we're talking about a 150 pound person, that's equal to eating between 37.5 to 75 pounds of food per day! And there are some species of birds, especially hummingbirds, that will eat the equivalent of their body weight (or more) in food every day. Even larger birds that eat a smaller percentage of their body weight still eat far more than most of us humans ever would in a day.
Wild Bird Club notes, "A chickadee may eat 35 percent of its weight in food each day while a blue jay eats about 10 percent of its weight and a common raven 4 percent. Those percentages may not sound like a lot compared to hummingbirds, but put it this way: a 150-pound person would need to eat over 52 pounds of food each day to eat like a chickadee, or 15 pounds to eat like a jay, or 6 pounds to eat like a raven."
So the next time someone tells you that you eat like a bird, you can contradict them with confidence.