It's called a Companion Cube for a reason: test subjects should become attached.
The difference with regular cubes is for measuring emotional responses and ethical choices made by the test subjects, and likely for general reasons of A/B testing, to compare results in test performance between the two different iterations of what is essentially the exact same thing.
Similarly with the turrets. Remember that Aperture Science Labs' mission, in Cave Johnson's words, was "throwing science at the wall and seeing what sticks", i.e. innovate at whatever cost, regardless of moral and ethics.
[T]hough why turrets would be made sentient to such a degree that they apparently understand why they're destroyed, I don't know.
Possibly because Aperture Science also didn't know, and figured they might as well find out :) The turrets might contain Morality Cores, in addition to or without conscience, or they might just produce random soundbites. They might be cruel experimental or harmless straightforward death machines, but they will impact the test subjects in some measurable way.
Apart from the possibility that GLaDOS is simply lying to get what she wants, and always undermining attempts at positive human behaviour (probably to keep the risk of being overthrown as low as possible), those remarks about the Companion Cube tell us more about the history of the test subjects, and GLaDOS' view of them, then about the Weighted Companion Cube itself:
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While it has been a faithful companion, your Companion Cube cannot accompany you through the rest of the test. If it could talk — and the Enrichment Center takes this opportunity to remind you that it cannot — it would tell you to go on without it because it would rather die in a fire than become a burden to you.
To me, this insinuates that test subjects - growing up in complete isolation save for an AI controlling their lives and making snarky comments - lack actual companionship, and grow attached to objects. The rest is just motivational nonsense, that these AIs are so good at spewing (cf. Wheatley).
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Although the euthanizing process is remarkably painful, eight out of ten Aperture Science engineers believe that the Companion Cube is most likely incapable of feeling much pain.
This tells us that even the Aperture Science engineers sometimes have difficulty separating reality from fiction, an indication that they're not much better off than the test subjects (at least, when it comes to oppression—less so when it comes to risking their lives). The staff were actually increasingly used as test subjects, as well, so who knows at which point these statistics (which may be false) were measured.
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Rest assured that an independent panel of ethicists has absolved the Enrichment Center, Aperture Science employees, and all test subjects of any moral responsibility for the Companion Cube euthanizing process.
Unless there is something living or intelligent inside the Companion Cubes, this is just drivel—and likely intended that way: it is an opaque attempt to clear the test subject's conscience, while more likely leading to the very question "is this wrong?". Reverse psychology.
That the Companion Cube reappears at the end credits of Portal can also be interpreted in many ways: it might be indestructible, or perhaps Chell has lost her mind, or there are simply lots of copies around. In Portal 2 we find out there are a lot (or at least several) of these Companion Cubes (Test Chamber 7: "We have warehouses full of the things"). I believe even in Portal you can already glimpse others behind the walls.
In the canonical comic Lab Rat, Rattmann is seen conversing with the Cube, but we know he was delusional, a schizophrenic suffering from hallucinations:
It's hard to dismiss all the insinuation, and it can't be proven the Cubes don't have an intelligence or sentience, but to me everything points in the direction of a ploy. For science.
