All About Me?
This tension has existed for a long time in the church - think about the notion of "personal savior" and individual salvation in addition to the Biblical stories about the whole people of Israel as the chosen people and membership in the tribes trumping (or at being least a major part of) any individual identity. Think about Jesus incorporating "the Gentiles" into his ministry and yet also the stories about individuals within those larger groups.
Paul was reminding the Corinthians, who possessed powerful spiritual gifts, that they were still part of a much larger body than just themselves. They were splitting the body - and in fact, not only were they not connected to the other Christian communities around them but they were splitting themselves into even smaller factions within their own community.
You can always tell what Paul is going to be up to in his letters by reading the greeting. You see that immediately in this letter: he describes the Corinthians as those who are to be called "the saints together with all those who in every place call on the name of our Lord Jesus Christ, both their Lord and ours." In other words, you're not just the Corinthian Church. You are also part of the whole Body of Christ.
So, we observe Lent (and indeed, all of the seasons of the church) both as individuals and as a body. We have individual practices and communal practices. Paul reminds us not to let our individuality overwhelm our corporate identity as members of a larger body, the Body of Christ.
The more things change, the more they stay the same.