Barbells are simply bigger and heavier dumbbells. However most dumbbells in the context the comment was referring to aren't heavy enough to actively engage the stabilizing muscles to any great effect.

The exercise also makes a big difference. Doing chest flys with dumbbells may feel like you are using a lot of stabilizers but actually you are no where near using as many as a free standing squat.

And of course you're using bigger muscles when doing more compound exercises so correspondingly the weight have to be bigger. I believe that's why in general most people would say barbells > dumbbells.

I would also add that there are plenty of compound dumbbell exercises. A compound lift is simply one that uses more than one joint.

Totally. You can do a clean and jerk with two dumbbells, for instance.

Yes, but I believe the stabilizer muscles (rotator cuff) can't continue to support the forces placed on them at a high enough weight. Generally, people "max out" stabilizers before muscles like pecs. So you plateau with dumbbells before barbells.

According to "starting strength", at some point you can't hold the dumbbells due to limitations in grip strength.