I'm not sure how much philosophical work has been done on the concept of solidarity, honestly. I think it might be useful, at least for me, to explain what I think of it. There's not a lot of point to this post beyond trying to explain something to my own satisfaction; feel free to skip if that sort of thing doesn't interest you.
Solidarity relates closely to group membership, obviously. You can't exhibit solidarity with yourself, unless you reconceive yourself as some sort of weird collective -- an army of one, if you like. Someone else needs to be involved for solidarity to even exist.
However, solidarity can't be exhibited to just any random person. There has to be some kind of connection between everyone. Membership in a group of some kind thus seems key. I wouldn't want to define "group" in any limited way, though. It's possible to exhibit solidarity in relation to a loose gathering of people with a common function, such as members of a job group, or with a common set of needs and experiences, such as members of an economic class.
Solidarity is quite similar to loyalty. It involves sticking close to the members of a group, supporting them in times of difficulty, standing with them even if it harms one's own interests. It also involves a diachronic perspective, focused not just on an immediate moment, but on a broad stretch of time. Like loyalty, exhibiting solidarity requires a historical commitment.
I tend to think of loyalty as personal, though. You can be loyal to a friend or an employer, but solidarity seems less personal, involving committment to something less personal, a group of people. (As the group's existence swings somewhat free from the people within it -- people can leave or enter a group without affecting the group's existence.)
Solidarity, to be an actual moral value, has to be connected to practical rationality. So, on the "rational" end, it will require involvement of one's intelligent/cognitive faculties. Someone who exhibits solidarity can't be oblivious to the nature of the group in question. Solidarity isn't commitment come what may. It requires an awareness of what the group is actually like, not what one would wish it be like. You can't exhibit solidarity to an ideal by sticking close to a group of people who only pay lip service to that ideal.
Similarly, solidarity will require deploying and developing one's knowledge about the collective and whatever defines it. Otherwise, it's a kind of blind faith.
On the "practical" end, solidarity has to have a connection to action. It's not a value that one simply possess, but one that must be used or demonstrated through one's behaviour. This can be as simple as public endorsement, or as complex as ongoing employment or activism.
So, centrally, solidarity requires an impersonal commitment to a somehow defined group, and is moral to the extent that it involves one's reason and manifests in action. Is that too thin? Possibly, but it does seem to distinguish solidarity from other key moral values.