Wolves lick each others wounds, for example.
Although the behavior evidenced by cetaceans is fascinating, it just isn't societally important enough to justify the push by the AAS to recognize them as fundamentally equivalent to people.
Of course, nothing in biology is that simple. The individuals could have some other use, such as navigational memory. The carers could be gaining reputation, i.e. they'll get more benefits from other members of the group. Or it could be an evolutionary 'accident': the injured or weak individual triggers a response that evolved to restore a productive member of the group, even if it won't work in that case.
You can make similar arguments about human altruism. But in some cases, the most simple explanation is a personal relationship between individuals and a sense of compassion.
I don't have strong feelings either way about the initiative in the linked article. But I'm fascinated by the questions of consciousness and how we assess it.
Not really. The Social constructs are beneficial to gene survival. If a woolly mammoth injures you, I take care of you because I have a social bond that works both ways. Prehistoric me does not necessarily understand if the injury is permanent, but our bond is there to assume that the help I'm giving you will be conferred upon me or my offspring at some point.
Love, cooperation, and compassion are (situationally) survival traits.
When you consider further that individual survival isn't as important as genetic survival, you can see that social behaviors are selected for in animals capable of exhibiting them to some degree.