Learn English – The state of being a member of something

Simply compound non- (negation) and singleton to get non-singleton, which will be readily comprehended by anyone who knows singleton. For example, in a hand of cards, you can refer to a non-singleton king and it will be understood that there is at least one other card of that suit in the hand.

In English, it is common to express meaning using compounds. In particular, there are many productive prefixes and suffixes which can be meaningfully compounded with a root word.

The catch is that almost every formal group of people has a "distinguished member". Any term implying a group of people with some structure will thus imply the existence of special roles within the group. So there hasn't been much need in English for such a specific word.

In line with that, the best I can suggest is the word "organization". (I considered "society", but the more general meanings of that word probably mean it's not as useful.) An organization implies that a group has some internal structure, and that structure in turn implies that some members will have special roles, without necessarily implying the existence of a hierarchy.

If this suggestion isn't useful, you should provide some context as to exactly why you need such a word.