It's good to want things.
That's what I tell my 11-year-old daughter when she says she wants an iPad for Christmas or a lock on her bedroom door or a convertible when she's 16. I think you should tell your boyfriend the same thing. It's a slightly less confrontational way of saying, "Tough titties, Paco.
Your guy certainly has a lot of opinions about your life, particularly for someone you've only dated a short time. The real question is, do you want to quit Instagram? It's unfortunate that your social networking makes him uncomfortable, but that's more his problem than your problem. If you like Instagram, Facebook and the like, if these activities give you pleasure, then I don't think it's fair of him to ask you to stop because it makes him uncomfortable. If he doesn't like Instagram, he doesn't have to use it. If your posts make him uneasy, he should stop reading them.
Now, if he objects to you talking about him or posting pictures of him on social networks, that's his right and you should honor that. You can offer to leave him out of your posts. I believe that is a fair compromise. I don't think you should give up something you enjoy and that isn't hurting him just because he doesn't like it. He's your boyfriend, not your dad.
One more thing: I understand that you want to keep things peaceful in your relationship, but if you agree to this, what will he demand of you next? It's a slippery slope. For a short-term relationship like yours, I think it's much more in your self-interest to stand your ground and do what you want. If he bails because he can't control you, you are better off, anyway.
Thanks for the question.