Could a person who hasn't had bariatric surgery lose weight by following the same type of diet? Could a person who hasn't had bariatric surgery lose weight by following the same type of diet?

You certainly could if you could handle eating such small amounts of food, but they will lose faster, because they can't digest even the small amount of food effectively enough to get all the calories out of it.
If you are serious about weight loss, and don't want surgery, let me urge you to go to your doctor at once, and get checked out to see if it's safe to start a sensible diet and exercise program. Find out if there is a thyroid problem for example, and if so, get that treated. Get a referral to a nutritionist who can give you a healthy, balanced diet to follow that is based on what YOU like to eat. Either that, or try Weight Watchers for their diet--also a sensible plan. Then, begin an exercise program. Your first goal is 30 minutes of cardio a day, every day. If you can't handle that right away, start with 10 or 20 minute walks. Ease into more, until after maybe eight weeks you are doing a 30 minute walk a day. Then, start lifting weights--even light weights will help. Join a gym to get some pointers on designing a workout, or get a book like "Strong Women Stay Slim". Initially you can use just some light dumb bells at home, and your own body weight for resistance training, as it's called. Eventually, you will be able to add more weight, or more repetitions and sets of repetitions. As you add muscle, you will burn more calories. As you burn more calories, you will lose weight faster. You will also look and feel better.