This particular verse is written in such a way as to be translated either "with" or "in": Our fellowship is with Christ, or our fellowship is in Christ. It can go either way. The case is both subjective and objective in I Corinthians 1:9.
Fellowship means "sharing," "communion with," "companionship with," or "association with." We have been called into an association—a companionship, a fellowship, a communion—with Christ. All these words are synonyms. The only difference might be the degree of the intimacy that is expressed. In addition, fellowship indicates people having things in common—they do things together because they share common interests. What we have in common is our love for Christ.
We are drawn to the brethren because of the common tie—the common love for the same Person. Even when we meet people in the church for the very first time, we do not feel as though they are perfect strangers to us because of that commonality. We recognize the spirit or attitude that emanates from them. It is almost something that we can feel or see because our senses seem to be attuned to it. This is why world travelers with the church say that they can go into another congregation and know that it is of the same Spirit as the one that they traveled from.
There is a bond or union between us because we love the same Person. To the Christian, then, Christ's friend is our friend. We are members of the same body. We are children in the same Family. We are soldiers in the same army. We are pilgrims on the same road. These same analogies are used many places in the Bible.