Sounds like you are setting a great example by having lots of healthy meal options at home. Congratulations!
To help answer your questions about your son, we need to get an idea of his potential health risk. To do this, docs will use the Body Mass Index percentile, just like we use the growth chart percentiles to give context to the height and weight measurements over time and relative to other kids.
Studies of docs relying just on their eyeballs to determine risk based on weight show that we don't do a good job of picking out kids at risk or potentially at risk as they grow up. So the first step is to make a diagnosis based on BMI percentile for age using the child's height, weight, gender, and age.
Increased risk of health problems is greatest for BMI percentiles greater than the 95%ile for age. There is some increased risk between 85th and 94th%iles as well. I tried to calculate your child's BMI percentile but I wonder if there is a typo in your child's height above. 28 3/4" is more like the height of a 9 month old. Maybe his height is 38 3/4"?
Using the taller height and guessing your son's age to be 3 yrs and 1 month yields a BMI percentile of 91. This is not the highest risk zone, but something to pay attention to over time since there is a risk of it climbing higher as he grows up.
Limiting portion sizes is a useful strategy to encourage healthy eating habits for a lifetime. Hungry kids can always ask for more.
Studies of adults show that portion size makes a big difference. The classic popcorn bucket study is a great example. People were given free popcorn at a movie theater. Some people got large buckets and some got small buckets. Guess which group ate more popcorn? The large bucket group.
Then the twist. What about giving away stale popcorn? Even when the popcorn wasn't tasty, the large bucket people still ate more.
I think that some (many?) kids, just like adults, are capable of overeating and portion size can be an unconscious trigger to eat more.
A typical pattern that I see in my practice is kids who are maybe just a little heavy as preschoolers, then have a steep climb in BMI percentile through early elementary school years to become obese. I don't know why that happens, but it happens fairly frequently. I think your doctor has probably seen this same phenomenon and was very astute in picking up on a small increase in BMI percentile. Offering some advice at this very early stage can hopefully prevent the BMI percentile from rising higher over the next few years.