You knew from the beginning that they were still friends. Just because it didn’t work out doesn’t mean they can’t be interested in what happens in each other’s lives. The generous and trusting thing you can do about her birthday is to add your good wishes to his and support him if he wants to take her out for a glass a wine before coming home to you. If your relationship is strong, periodic contact with the ex isn’t going to matter.
The most important factor in a love relationship is mutual trust. If you can’t bring that to the relationship, when your boyfriend clearly deserves it, all the emotional aspects of love won’t make up for it. If you keep believing you are in a contest for his love, you will lose. At some point, he won’t be able to tolerate always being under suspicion.
No, I wouldn’t bring it up to him any more. From all you said, it’s your issue. Work on yourself. If you can’t relax, it might be helpful to see a therapist to get to the root of your anxieties and learn how to manage them. I hope you will. It sounds like the guy is worth it.
I wish you well.