He is getting all the nutrients he needs and your doctor is correct. Also, kids go through phases. Once he gets tired of bananas he'll be ready to experiment with something else, then he'll only eat avocado for a month, thats just how it goes. You said "13 month old doesn't want to eat". This is untrue. He does want to eat- he wants to eat yummy breastmilk and healthy yogurt and bananas. If your friends are scaring you don't listen to them.
Trust your doctor, those guys tend to be pretty smart. You are doing great my breast-feeding past a year and I applaud you. My son had this exact diet at 13 months and weighed less. If you push it he may resist more. I used to breathed and then do a breast milk smoothy with the yogurt and banana. Try that and then change it up to a different kind of smoothy just to get a new taste in there in a familiar way, the smoothy looks the same but tastes different, now you've introduced a new taste in a non threatening way. If he likes the taste of an avocado or strawberry smoothy maybe next time put the sliced banana on his plate with a sliced strawberry next to it- he might be intrigued.
Reading picture books with different foods and people eating them is great. Also, try eating with him. If he's eating banana and your eating a strawberry, ask if he'd like a bite. Most kids want to do what we're doing.
Lastly, try to relax and enjoy your sweet baby!