You can have something give you a feeling without you belonging to their organization.It really doesn't. If you've never been a member of an Anglican church, how can you know what "it feels like"?
The Episcopalian Church USA website defines the Anglican Communion as being those churches in communion with Canterbury. So if you leave communion with that particular see or get excommunicated by them, you no longer belong to the Anglican Communion that the ECUSA does. Canterbury, like the See of Rome among Catholics, thus becomes the necessary point of unity for their communion.
In contrast, the Orthodox Church worldwide does not have a single see that churches must be in communion with in order to belong to our communion. If one See excommunicated another, it does not automatically follow by virtue of the location of one that the other is excluded from the whole Orthodox Communion. The same principle is true in Lutheranism AFAIK that I just laid out in Orthodoxy.