My son is very similar, but 10 lbs lighter. He's always eating and working out. He just had his annual checkup and we talked with his doctor about it. She thinks he still has an inch or so to grow but should start being able to put weight on. His metabolism is so high she said it has been basically impossible for someone like him to add any significant weight for the past couple of years.
My son drinks a ton of whole chocolate milk, snacks on protein bars, and also eats a couple of uncrustables throughout the day for snacks. Now that he's driving he will also stop for a jersey Mike's sub when he has time.