What a tough situation! Somehow your son IS obsessed with his special friend. He loves her, which is great. But he also feels that he needs to be constantly connecting with her, or he doesn't feel good inside. He's so afraid of losing her that he needs to "control" her. That's a problem for both of them.
1. Start setting up play opportunities outside of childcare for your son to play with kids who are at childcare with him. If he can make other friends who are at childcare, that will help him at childcare when his special friend wants to play with others. Choose both boys and girls, so that at childcare he has options.
2. Get photos of all the kids and put them in a little booklet. Read the book with him and talk about all the kids. "That's Henry...what does Henry like to do? Blocks? Do you ever play blocks with Henry? The block corner is so much fun!.....And that's Sara. She loves the water table. Do you love the water table? I bet Sara would love for you to play with her." Your goal is to open his mind to the fact that there is a whole class of kids, and some of them are pretty cool and would love to play with him.
3. Enlist the teachers to help with this situation. They should be able to talk with the class about the rule "You can't say someone can't play" which is a fairly universal rule at preschools, meant to preclude cliches and one-on-one exclusivity. Your son can understand that this is the rule, and everyone is allowed to join in play with his friend. Also, though, ask the teachers if they can support your son emotionally when he gets upset about the situation. Maybe they can intervene and invite him to do something else. When you're feeling heartbroken and dissed because your crush is laughing with others, it can really help to have someone else pull you into another activity.
4. Help your son work out the feelings that are causing him to feel so jealous about sharing his friend. Begin by playing with him about it. Say "Let's play childcare!" You be the girl, and he can be him. Be very silly with this game, to get him giggling. Play a bit, then tell him that you love him but you also have other friends that you want to play with. Ask him if you can invite other friends to play also. (These friends can be stuffed animals.) Maybe he will say NO! and want to throw the stuffed animals out of the room. That's allowed. This is just play, after all, and play is how kids work out their feelings. As long as he's giggling, he's letting out his built-up anxiety about having to share his friend.
5. Help your son get in touch with those raw feelings of desperate need. Right now, your son feels like he NEEDS his friend and if he loses her he will not survive. That scares him terribly. So he gets very afraid when she plays with anyone else because he is afraid of losing her. If you can help him get in touch with that fear, it will have much less power over him. After playing the fun childcare game a few times with lots of giggling, make it more serious. Be the girl, and tell him that you love him but you also have other friends that you want to play with, that you cannot always be with him. Be very compassionate, so he feels safe, and hopefully, he will cry about this. If you stay in your role as the girl ("I'm sorry it is so hard for you, but I like other friends, too") he may really feel his need, fear and loss, and let it all out.
6. Let your son have the experience of being the one who wants to go, while someone else clutches at him. Sometime when he doesn't expect it, grab him onto your lap, and when he tries to get up, plead with him to stay with you, because you need him so much and can't do without him. Ham it up to get him laughing. Hug and kiss him. Let him feel like the mature, powerful, confident one while you are the needy, possessive one. Finally let him go, saying "It's ok...I know we will always love each other and you will always come back."
7. Let him process with you his feelings about what's happening at school. When he tells you he only wants to play with his special friend, don't correct him. Empathize: "You really only want to play with her....but she wanted to dress up with the other girls...and then you didn't know what to do...you must have felt so bad...." Again, your goal is to surface these feelings so he can release them.
8. Keep getting together with his friend's family. Don't think you're fanning the flames of his obsession. After all, he sees her at childcare two days a week. Depriving him of contact with her outside of childcare, but keeping them together in childcare, worsens the childcare situation by making him feel more deprived. It's good for him to feel like he gets plenty of opportunities to play with her.
9. Make him a book about his life. Include this friendship with a big enough role to show how important it is to him, but small enough to keep it in perspective. Right now, your little guy is in misery, worried about losing the key to his happiness. You are trying to give him a positive story about it to help him understand what's happening. Create a book -- something simple that is made of pages in a looseleaf binder. Print out photos and add captions. Start with you and your partner wanting him so badly (and of course any older siblings.) Show him as a newborn. How happy everyone was. Then introduce his "first friend" who was born the previous day. Say how they always loved being together. Add other photos of his life, his family, trips you have taken, him having fun with other children. Include his special friend a lot, but not in every spread. For school, show her in only one photo. Have a photo of him with the teacher, and him on the slide, and him with another kid. For one of the photos of him with her, use a caption like "Sometimes X wanted to play with other kids and Y felt a bit jealous. But he knew they would always be good friends." End the book with a high point, like his birthday party, "And then Y was three years old!" with her in the photo if you have one. That implies she's around to stay. The goal of the book, again, is to acknowledge the relationship and help him feel safe in it, but to show that his life is larger than this one relationship. It will also get the message across that sometimes he feels jealous because she wants to play with other kids, but that's ok, and they will always be friends.
I'm hopeful that these ideas will help your son work through his feelings of fear and need, so he doesn't feel like his happiness and wholeness depend on having his friend's attention. But if it doesn't, you need a graceful exit strategy for him. Get on a waiting list to move your son to another childcare center, just in case. But be aware that moving him won't change his feelings. In fact, he will be broken hearted, and will go through a grieving process. So in my opinion, it's better for him to learn to manage these feelings, so that he isn't at their mercy, rather than simply separating the children. But you do need the back up of a transfer option just in case.
Good luck, and please do let me know how things unfold.