How often have we heard the expression and maybe even used it ourselves: "I'm proud to have been born a ('Jew', .'American', etc.)" But if we stop and think about it, that's a baseless perspective. For example (setting aside the complex issue of ethnic Jewry and discounting the maternal lineage tradition) in the case of adherents of the Jewish religion, no one is "born" a Jew, or for that matter as a member of any particular faith. We come into this world as natural atheists but are indoctrinated by our families and culture into a set of theistic beliefs and values—usually beginning in the most impressionable period of our lives: early childhood. In other words, the religious identity that we come to accept for ourselves is the result of an accident of birth.
Another problem with having pride in one's religious beliefs or background, nationality, race, etc. is that this mindset can lead to feelings of superiority over other people who we deem as inferior because they are not "one of us". On the other hand, feelings of shame or inadequacy over one's birth status are likewise unproductive. Again, no one asked to be born into his or her particular demographic setting. But those who are particularly unhappy with their lot, see no chance of improvement in their futures, and have the opportunity to make a change (conversion,immigration, etc.) then they should seriously consider taking that step and act accordingly, which of course many such people do.
On the other hand for those who are content with whatever group that they happen to have been part of since birth, there's nothing wrong with being happy that things turned out this way. However, pride itself should come as the result of an accomplishment or achievement, not from a random act of fortune.