Morrigan can definitely care and likely does a lot more by the time she's in DAI, she was just very much a "people can be expendable if it helps our quest" kind of character in DAO in that she approved of stepping over or disregarding others to fit your goals. Much of this was probably because she was raised so isolated, which Solas could be also, but I just hope he doesn't follow exactly in Morrigan's footsteps in that way for the sake of not repeating personalities. I didn't go nearly into how I overall see the character so our views of her could possibly be similar! We definitely differ on Elthina, though.
As for Solas being that character who understands you not fully supporting his possible causes or possible connection to the plight of his people, I kind of like the idea but then I kind of don't. It'd be nice if everyone thought so logically about things but it's not really the way most people think and it comes off as slightly unrealistic. When you really care about something, you're going to be biased towards that and feel that it is at least partially a priority. However, all of the above only applies IF he feels connected and responsible for his people, which we don't know. He could easily not heavily identify with the mage cause or the Dalish/elf cause. That's all up in the air!
As I said above... whether or not he feels connected to anyone, he's already inherently involved by being an elf and a mage. While I'm sure the Veil Tear will take precedence over most if not all of the Inquisitor's companions, I would find it suspect if he didn't care at all about mages or elves. I could see it more if he were, for instance, a Tevinter human. Why would they care about the mages of Orlais and Ferelden? Granted, there would be reasons to care, but mostly it wouldn't be their fight. And if he were human, even if he were elf-blooded, his main concern about the elves would be an uprising.
Since he is, however, an elf and an apostate mage, I'd think it would matter to him whether or not he should expect to continue to be hunted by templars, and also whether or not he might be expected to join some of the elven factions. I don't know to what degree that is. Fenris didn't really identify as an elf, for instance, and yet, he was an elf so he couldn't exactly get away from that. And I think he knew it--knew that was part of the reason he'd been treated as he had. If the character's an elf in Orlais or Ferelden, they definitely know that it matters. Even the Dalish know (and are bitter about it, too), even though the oppression of the elves affects them differently.
The question is really, " to what degree do these issues matter to him?" And then you could go on to ask, "How do these issues matter to him?" For instance, as an apostate, they could go the route of his being upset by the mage/templar conflict not because he doesn't like the oppression of mages, but because it means that templars are working harder to rout out any apostates, putting him in greater danger than he was before. Even if he thinks most mages should be in Circles and strictly controlled, or even if he thinks all magi should be completely free, the templars' intensified searches would probably force him to react, unless he's actually Dalish--which is unlikely, what with his lack of tattoos.
And as an elf... TME spoilers ahead: