As you yourself have rightly learned and pointed out, it is not anything against Muslims. Everybody wants to be surrounded by their own community. They will not let you in even if you are a Hindu and not a Gujarathi etc. I don't want to judge this. I just want to say that it is no different from you moving to a Muslim locality after your worst experience. Even others must have had similar experiences.
You have asked "Why do we call ourselves Indians and want to take pride in our multi-cultural, multi-religious land when we cannot stand our neighbours who belong to different religions and cultures?"
First part of your question "Why do we call ourselves Indians", tell me brother what you want to be called as? Whatever are the options, just tell me, and we will examine the options.
I just wish you asked a different question "Why do we Indians want to take pride in our multi-cultural, multi-religious land when we cannot stand our neighbours who belong to different religions and cultures?"
It is not the same question asked differently, believe me, it's altogether a different question. And a valid one too.