I think somewhere along the way you added some meaning to "share" that it doesn't have.
We talk about "shares" pretty commonly here because a NAS without a share is not a useful NAS. A share is nothing more than a way to access some data remotely. Data might be shared using CIFS/SMB, FTP, AFP, NFS, or any other way. FreeNAS has lots of ways of sharing data.
A dataset is a feature of ZFS. A partition is kind of analogous, but don't think about it that way. What matters is that a dataset is the smallest organizational unit in ZFS that you can apply ZFS properties to. You can snapshot datasets, you can set compression/encryption on a dataset. A dataset can even have child datasets. But you can't do that to half a dataset, like a folder in a dataset, for example.
A dataset appears like a folder in the file hierarchy. And you can manipulate it like a folder. But it's much more powerful than a folder.
Tying that together, you usually share a dataset (otherwise, why have the dataset (ignoring jails and the root dataset)). So, when someone says "My media share is acting up", they usually mean that they created a "media" dataset, then they shared that dataset, and now they have a problem with either their media dataset, the service they are using to share that dataset, or both.