My DD was very high needs and needed to be held all the time. This was quite a shock since my son was so laid back. What helped me was once she was asleep I'd hang with her until she was really really out. Like deep sleep to the point if I rearranged her arms or legs she didn't even notice. Then I'd make sure she was swaddled and tuck a dirty shirt around her. Usually whatever shirt I had worn the day before. She could still smell me so she was ok.
It is a survival instinct for him to want his mom 24/7. Even if he was fed formula, he'd still want her. She is his whole world. The only thing that is familiar to him is her smell, her voice, her heartbeat.
And yes it does get better. Usually around 6-9 weeks, but it's different for everyone. Just roll with it. He'll never be this little again. You'll blink and it'll be his first birthday. I know right now it's all sleep deprivation and worry but you'll find a new normal.
Oh and make sure to give her time to shower and eat. Even if he is unhappy in your arms, he is still being comforted. Maybe not in his preferred method of mom, but he'll learn that daddy has his own set of awesome qualities. There will be a learning curve for both of you, but you two will find your groove together as well.