We have a very different experience, but my sons Muslim friends live in Norway. Maybe it is different.
"Calvin", 12, is Lebanese. He is not allowed to go to birthday parties. My sons friends classmates got around this by calling it Summer parites etc, and opted to forego presents. They ate pizza, Cake, etc, and played. Calvin invited his friends to Winter parties at the time of his birthday, and got "Happy Winter Cards" from his friends.
"Yassin", 9, is from Morocco. His family let him attend birthday parties and celebrate his own. His mum would frequently invite me in for cups of tea and snacks, and we loved discussing religion together, finding we had more in common than dividing us.
If the boys came to play I would cook chicken or fish for dinner, if we went for a picnic I would pack sandwiches with jam or chicken spreads. Chicken burger at burger king as a treat, etc. We took the boys skiing and mountain hiking, which they loved.
I find both families had found ways of integrating yet keep their religion and their traditions their own way, and were happy to send their children (sons and daughters) to play at the homes of non-muslim familes, and happy to invite back.
My oldest in particular loved to play at Calvins house, as his dad is a chef, and would cook delicious food.
I think it depends on the families.
I also think the English are the same, in my experience they are happy to keep to themselves and not too keen to establish new friends with foreigners. (The adults) There are exceptions of course!