Some kids are shy/quiet and you could engage with him by inviting him to participate in some fun activities. Reading, playing games, drawing, painting, Lego, cars, making forts, cooking. If you go and pick up some of his toys and start playing with them, he may decide to join you. It may not happen straight away but it is worth trying to do things he likes.

Good on you for trying to bond. Maybe suggest some activities that you can all do together, like playing ball, playing at the park. Also make sure he has time by himself with his dad. Good luck

You’re already doing well being sensitive, thinking about what HE is like. Take it slow, he will likely take a while to warm up to you. What does he like? Maybe start in house so baking (send some back to the mum if there’s a good relationship. He may like baking for her). Go to a park once he gets comfortable. Just small frequent activities and chats will build a relationship. Like throwing a ball together. Get him to help you with a chore and praise/encourage/thank him. Good luck!